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Let $V^{\otimes n}$ be the $n$-fold tensor product of a vector space $V.$ Following I. Schur we consider the action of the symmetric group $S_n$ on $V^{\otimes n}$ by permuting coordinates. In the `super' ($\Bbb Z_2$ graded) case $V=V_0\oplus V_1,$ a $\pm$ sign is added [BR]. These actions give rise to the corresponding Schur algebras S$(S_n,V).$ Here S$(S_n,V)$ is compared with S$(A_n,V),$ the Schur algebra corresponding to the alternating subgroup $A_n\subset S_n .$ While in the `classical' (signless) case these two Schur algebras are the same for $n$ large enough, it is proved that in the `super' case where $\dim V_0=\dim V_1, $ S$(A_n,V)$ is isomorphic to the crossed-product algebra S$(A_n,V)\cong$ S$(S_n,V)\times\Bbb Z_2 .$ | Double Centralizing Theorems for the Alternating Groups | 13,600 |
The 321,hexagon-avoiding (321-hex) permutations were introduced and studied by Billey and Warrington in as a class of elements of S_n whose Kazhdan-Lusztig and Poincare polynomials and the singular loci of whose Schubert varieties have certain fairly simple and explicit descriptions. This paper provides a 7-term linear recurrence relation leading to an explicit enumeration of the 321-hex permutations. A complete description of the corresponding generating tree is obtained as a by-product of enumeration techniques used in the paper, including Schensted's 321-subsequences decomposition, a 5-parameter generating function and the symmetries of the octagonal patterns avoided by the 321-hex permutations. | Explicit Enumeration of 321,Hexagon-Avoiding Permutations | 13,601 |
We derive new combinatorial identities which may be viewed as multivariate analogs of summation formulas for hypergeometric series. As in the previous paper [Re], we start with probability distributions on the space of the infinite Young tableaux. Then we calculate the probability that the entry of a random tableau at a given box equals n=1,2,.... Summing these probabilities over n and equating the result to 1 we get a nontrivial identity. Our choice for the initial distributions is motivated by the recent work on harmonic analysis on the infinite symmetric group and related topics. | Random Young Tableaux and Combinatorial Identities | 13,602 |
We extend work of McKay, Morse, and Wilf by giving exact formulas and asymptotic formulas for the number of skew Young tableaux T in two situations: (1) the "inside shape" and total number of cells of T are fixed, and (2) the inside shape of T is fixed. The proofs use the theory of symmetric functions and estimates for irreducible characters of the symmetric group due to Thoma, Vershik, Kerov, and Biane. | On the Enumeration of Skew Young Tableaux | 13,603 |
We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to the class P substitutions of Hof-Knill-Simon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block-)complexity. | Palindrome complexity | 13,604 |
Exponentiating the hypergeometric series gives a recursion relation for integer sequences which are generalizations of conventional Bell numbers. The corresponding associated Stirling numbers of the second kind are also generated and investigated. For the lowest order generalisation, one can give a combinatorial interpretation of these 'Bell' numbers, and of some Stirling numbers associated with them. We also consider these analogues of Bell numbers in the case of restricted partitions. | Extended Bell and Stirling numbers from hypergeometric exponentiation | 13,605 |
In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids. | Quadratic Orlik-Solomon algebras of graphic matroids | 13,606 |
The Ramanujan polynomials were introduced by Ramanujan in his study of power series inversions. In an approach to the Cayley formula on the number of trees, Shor discovers a refined recurrence relation in terms of the number of improper edges, without realizing the connection to the Ramanujan polynomials. On the other hand, Dumont and Ramamonjisoa independently take the grammatical approach to a sequence associated with the Ramanujan polynomials and have reached the same conclusion as Shor's. It was a coincidence for Zeng to realize that the Shor polynomials turn out to be the Ramanujan polynomials through an explicit substitution of parameters. Shor also discovers a recursion of Ramanujan polynomials which is equivalent to the Berndt-Evans-Wilson recursion under the substitution of Zeng, and asks for a combinatorial interpretation. The objective of this paper is to present a bijection for the Shor recursion, or and Berndt-Evans-Wilson recursion, answering the question of Shor. Such a bijection also leads to a combinatorial interpretation of the recurrence relation originally given by Ramanujan. | Bijections behind the Ramanujan Polynomials | 13,607 |
Babson and Steingr\`imsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1,2) or (2,1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. In the present paper we give a complete solution for the number of permutations avoiding a pair of patterns of type (1,2) or (2,1). We also conjecture the number of permutations avoiding the patterns in any set of three or more such patterns. | Enumerating permutations avoiding a pair of Babson-Steingrimsson
patterns | 13,608 |
Let m and n be integers, $2 \leq m \leq n$. An m by n array consists of mn cells, arranged in m rows and n columns, and each cell contains exactly one symbol. A transversal of an array consists of m cells, one from each row and no two from the same column. A latin transversal is a transversal in which no symbol appears more than once. We will establish a sufficient condition that a 3 by n array has a latin transversal. | Latin transversals of rectangular arrays | 13,609 |
A bijection is given between fixed point free involutions of $\{1,2,...,2N\}$ with maximum decreasing subsequence size $2p$ and two classes of vicious (non-intersecting) random walker configurations confined to the half line lattice points $l \ge 1$. In one class of walker configurations the maximum displacement of the right most walker is $p$. Because the scaled distribution of the maximum decreasing subsequence size is known to be in the soft edge GOE (random real symmetric matrices) universality class, the same holds true for the scaled distribution of the maximum displacement of the right most walker. | Random walks and random fixed-point free involutions | 13,610 |
Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in S_n which contain a given permutation \tau in S_k as a subsequence; this number depends on the patterns of the first j values of \tau for 1<=j<=k. We then use this to define a partition of S_k, analogous to Wilf-classes in the study of pattern avoidance, and examine properties of this equivalence. In the process, we show that a permutation \tau_1...\tau_k is layered iff, for 1<=j<=k, the pattern of \tau_1...\tau_j is an involution. We also obtain a result of Sagan and Stanley counting the standard Young tableaux of size $n$ which contain a fixed tableau of size $k$ as a subtableau. | Subsequence containment by involutions | 13,611 |
A critical set in an n x n array is a set C of given entries, such that there exists a unique extension of C to an n x n Latin square and no proper subset of C has this property. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n) <= n^2 - n. Here we show that lcs(n) <= n^2-3n+3. | A new bound on the size of the largest critical set in a Latin square | 13,612 |
We study generating functions for the number of permutations in $S_n$ subject to set of restrictions. One of the restrictions belongs to $S_3$, while the others to $S_k$. It turns out that in a large variety of cases the answer can be expressed via continued fractions, and Chebyshev polynomials of the second kind. | Restricted set of patterns, continued fractions, and Chebyshev
polynomials | 13,613 |
We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in enumerating matchings of planar graphs, up to matrix operations on their rows and columns. If such a matrix is defined over a principal ideal domain, this is equivalent to considering its Smith normal form or its cokernel. Many variations of the enumeration methods result in equivalent matrices. In particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus matrices. We apply these ideas to plane partitions and related planar of tilings. We list a number of conjectures, supported by experiments in Maple, about the forms of matrices associated to enumerations of plane partitions and other lozenge tilings of planar regions and their symmetry classes. We focus on the case where the enumerations are round or $q$-round, and we conjecture that cokernels remain round or $q$-round for related ``impossible enumerations'' in which there are no tilings. Our conjectures provide a new view of the topic of enumerating symmetry classes of plane partitions and their generalizations. In particular we conjecture that a $q$-specialization of a Jacobi-Trudi matrix has a Smith normal form. If so it could be an interesting structure associated to the corresponding irreducible representation of $\SL(n,\C)$. Finally we find, with proof, the normal form of the matrix that appears in the enumeration of domino tilings of an Aztec diamond. | Kasteleyn cokernels | 13,614 |
The distribution of a given sequence in the set of all sequences with n ones and m = M - n zeros are found by relating the problem to the partitions of a natural number in m natural summands, taking into account the order. The formulas obtained have many applications both in Physics and Mathematics. Examples discussed in the present paper are: non Markovian chains, partition functions of binary alloys and Ising magnets, generalized Kaplansky lemma, generalized Fibonacci numbers and a general expansion of \sum_{h=0}^{m} h^{r} {\binom{m}{h}}^{2} in terms of the Stirling numbers of second kind. | Structure of Binary Sequences | 13,615 |
For a fixed positive integer n, let S_n denote the symmetric group of n! permutations on n symbols, and let maj(sigma) denote the major index of a permutation sigma. For positive integers k<m not greater than n and non-negative integers i and j, we give enumerative formulas for the cardinality of the set of permutations sigma in S_n with maj(sigma) congruent to i mod k and maj(sigma^(-1)) congruent to j mod m. When m divides n-1 and k divides n, we show that for all i,j, this cardinality equals (n!)/(km). | On counting permutations by pairs of congruence classes of major index | 13,616 |
A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected set partitions respectively. The proof relies on Moebius inversion on the partition lattice. | Free cumulants and enumeration of connected partitions | 13,617 |
A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants. | Cumulants, lattice paths, and orthogonal polynomials | 13,618 |
Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type $(1,2)$ or $(2,1)$. For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. With respect to being equidistributed there are three different classes of patterns of type $(1,2)$ or $(2,1)$. We present a recursion for the number of permutations containing exactly one occurrence of a pattern of the first or the second of the aforementioned classes, and we also find an ordinary generating function for these numbers. We prove these results both combinatorially and analytically. Finally, we give the distribution of any pattern of the third class in the form of a continued fraction, and we also give explicit formulas for the number of permutations containing exactly $r$ occurrences of a pattern of the third class when $r\in\{1,2,3\}$. | Counting occurrences of a pattern of type (1,2) or (2,1) in permutations | 13,619 |
In [BS] Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let $f_{\tau;r}(n)$ be the number of $1\mn3\mn2$-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $\tau$, where $\tau$ a generalized pattern on $k$ letters. Let $F_{\tau;r}(x)$ and $F_\tau(x,y)$ be the generating functions defined by $F_{\tau;r}(x)=\sum_{n\geq0} f_{\tau;r}(n)x^n$ and $F_\tau(x,y)=\sum_{r\geq0}F_{\tau;r}(x)y^r$. We find an explicit expression for $F_\tau(x,y)$ in the form of a continued fraction for where $\tau$ given as a generalized pattern; $\tau=12\mn3\mn...\mn k$, $\tau=21\mn3\mn...\mn k$, $\tau=123... k$, or $\tau=k... 321$. In particularly, we find $F_\tau(x,y)$ for any $\tau$ generalized pattern of length 3. This allows us to express $F_{\tau;r}(x)$ via Chebyshev polynomials of the second kind, and continued fractions. | Continued fractions and generalized patterns | 13,620 |
Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on $n$ letters avoiding $1-3-2$ (or containing $1-3-2$ exactly once) and an arbitrary generalized pattern $\tau$ on $k$ letters, or containing $\tau$ exactly once. In several cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind, and generating function of Motzkin numbers. | restricted 1-3-2 permutations and generalized patterns | 13,621 |
Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following \cite{BCS}, let $e_k\pi$ (respectively; $f_k\pi$) be the number of the occurrences of the generalized pattern $12\mn3\mn...\mn k$ (respectively; $21\mn3\mn...\mn k$) in $\pi$. In the present note, we study the distribution of the statistics $e_k\pi$ and $f_k\pi$ in a permutation avoiding the classical pattern $1\mn3\mn2$. Also we present an applications, which relates the Narayana numbers, Catalan numbers, and increasing subsequences, to permutations avoiding the classical pattern $1\mn3\mn2$ according to a given statistics on $e_k\pi$, or on $f_k\pi$. | Continued fractions, statistics, and generalized patterns | 13,622 |
We find generating functions for the number of words avoiding certain patterns or sets of patterns on at most 2 distinct letters and determine which of them are equally avoided. We also find the exact number of words avoiding certain patterns and provide bijective proofs for the resulting formulas. | Words restricted by patterns with at most 2 distinct letters | 13,623 |
The vertex set of the kth cartesian power of a directed cycle of length m can be naturally identified with the set of k-tuples of integers modulo m. For any two vertices v and w of this graph, it is easy to see that if there is a hamiltonian path from v to w, then the sum of the coordinates of v is congruent, modulo m, to one more than the sum of the coordinates of w. We prove the converse, unless k = 2 and m is odd. | Hamiltonian Paths in Cartesian Powers of Directed Cycles | 13,624 |
A composition of an integer is called Carlitz if adjacent parts are different. Several characteristics of random Carlitz compsitions have been studied recently by Knopfmacher and Prodinger. We complement their work by establishing the asymptotics of the average number of distinct part sizes in a random Carlitz composition. | Average number of distinct part sizes in a random Carlitz composition | 13,625 |
Wilf posed the following problem: determine asymptotically as $n\to\infty$ the probability that a randomly chosen part size in a randomly chosen composition of n has multiplicity m. One solution of this problem was given by Hitczenko and Savage. In this paper, we study this question using the techniques of generating functions and singularity analysis. | A generatingfunctionology approach to a problem of Wilf | 13,626 |
This note reports on the number of s-partitions of a natural number n. In an s-partition each cell has the form $2^k-1$ for some integer k. Such partitions have potential applications in cryptography, specifically in distributed computations of the form $a^n$ mod m. The main contribution of this paper is a correction to the upper bound on the number of s-partitions presented by Bhatt. We will give a precise asymptotics for the number of such partitions for a given integer n. | S-partitions | 13,627 |
Let $P_r(n)$ be the set of partitions of n with non negative rth differences. Let $\lambda$ be a partition chosen uniformly at random among the set $P_r(n)$. Let $d(\lambda)$ be a positive rth difference chosen uniformly at random in $\lambda$. The aim of this work is to show that for every $m\ge 1$, the probability that $d(\lambda)\ge m$ approaches $m^{-1/r}$ as $n\to\infty$. To prove this result we use bijective, asymptotic/analytic, and probabilistic combinatorics. | Random partitions with non negative rth differences | 13,628 |
We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance of the number of (possibly overlapping) ordered labeled subgraphs of a labeled graph as a function of its randomness deficiency (how far it falls short of the maximum possible Kolmogorov complexity) and (ii) a new elementary proof for the number of unlabeled graphs. | Kolmogorov Random Graphs and the Incompressibility Method | 13,629 |
Let S be a set of 2n+1 points in the plane such that no three are collinear and no four are concyclic. A circle will be called point-splitting if it has 3 points of S on its circumference, n-1 points in its interior and n-1 in its exterior. We show the surprising property that S always has exactly n^2 point- splitting circles, and prove a more general result. | The number of point-splitting circles | 13,630 |
We first show that the tilings of a general domain form a lattice which we then undertake to decompose and generate without any redundance. To this end, we study extensively the relatively simple case of hexagons and their deformations. We show that general domains can be broken up into hexagon-like parts. Finally we give an algorithm to generate exactly once every element in the lattice of the tilings of a general domain. | An algorithm to generate exactly once every tiling with lozenges of a
domain | 13,631 |
In 1988, Kalai extended a construction of Billera and Lee to produce many triangulated (d-1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack, he derived that for every dimension d>=5, most of these (d-1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal. Moreover, we can now give a shorter proof of Hebble & Lee's 2000 result that the dual graphs of these 4-polytopes are Hamiltonian. Therefore, the polars of these Kalai polytopes yield another family supporting Barnette's conjecture that all simple 4-polytopes admit a Hamiltonian circuit. | Kalai's squeezed 3-spheres are polytopal | 13,632 |
The problem of counting tilings of a plane region using specified tiles can often be recast as the problem of counting (perfect) matchings of some subgraph of an Aztec diamond graph A_n, or more generally calculating the sum of the weights of all the matchings, where the weight of a matching is equal to the product of the (pre-assigned) weights of the constituent edges (assumed to be non-negative). This article presents efficient algorithms that work in this context to solve three problems: finding the sum of the weights of the matchings of a weighted Aztec diamond graph A_n; computing the probability that a randomly-chosen matching of A_n will include a particular edge (where the probability of a matching is proportional to its weight); and generating a matching of A_n at random. The first of these algorithms is equivalent to a special case of Mihai Ciucu's cellular complementation algorithm and can be used to solve many of the same problems. The second of the three algorithms is a generalization of not-yet-published work of Alexandru Ionescu, and can be employed to prove an identity governing a three-variable generating function whose coefficients are all the edge-inclusion probabilities; this formula has been used as the basis for asymptotic formulas for these probabilities, but a proof of the generating function identity has not hitherto been published. The third of the three algorithms is a generalization of the domino-shuffling algorithm described by Elkies, Kuperberg, Larsen and Propp; it enables one to generate random ``diabolo-tilings of fortresses'' and thereby to make intriguing inferences about their asymptotic behavior. | Generalized domino-shuffling | 13,633 |
Steingrimsson has recently introduced a partition analogue of Foata-Zeilberger's mak statistic for permutations and conjectured that its generating function is equal to the classical q-Stirling numbers of second kind. In this paper we prove a generalization of Steingrimsson's conjecture. | Nouvelles statistiques de partitions pour les q-nombres de Stirling de
seconde espece | 13,634 |
We show that a finite graded lattice of rank n is supersolvable if and only if it has an EL-labeling where the labels along any maximal chain form a permutation. We call such a labeling an S_n EL-labeling and we consider finite graded posets of rank n with unique top and bottom elements that have an S_n EL-labeling. We describe a type A 0-Hecke algebra action on the maximal chains of such posets. This action is local and gives a representation of these Hecke algebras whose character has characteristic that is closely related to Ehrenborg's flag quasi-symmetric function. We ask what other classes of posets have such an action and in particular we show that finite graded lattices of rank n have such an action if and only if they have an S_n EL-labeling. | EL-labelings, Supersolvability and 0-Hecke Algebra Actions on Posets | 13,635 |
Let P be a d-dimensional lattice polytope. We show that there exists a natural number c_d, only depending on d, such that the multiples cP have a unimodular cover for every natural number c >= c_d. Actually, a subexponential upper bound for c_d is provided, together with an analogous result for unimodular covers of rational cones. | Unimodular covers of multiples of polytopes | 13,636 |
We consider a filtration of the symmetric function space given by $\Lambda^{(k)}_t$, the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than $k$. We introduce symmetric functions called the $k$-Schur functions, providing an analog for the Schur functions in the subspaces $\Lambda^{(k)}_t$. We prove several properties for the $k$-Schur functions including that they form a basis for these subspaces that reduces to the Schur basis when $k$ is large. We also show that the connection coefficients for the $k$-Schur function basis with the Macdonald polynomials belonging to $\Lambda^{(k)}_t$ are polynomials in $q$ and $t$ with integral coefficients. In fact, we conjecture that these integral coefficients are actually positive, and give several other conjectures generalizing Schur function theory. | Schur function analogs for a filtration of the symmetric function space | 13,637 |
We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act simply on the k-Schur functions, thus leading to a concept of irreducibility for these functions. | Schur function identities, their t-analogs, and k-Schur irreducibility | 13,638 |
A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice cells. The corresponding lattice diagram determinant is \Delta_L(X;Y)=\det \| x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space M_L is the space spanned by all partial derivatives of \Delta_L(X;Y). The ``shift operators'', which are particular partial symmetric derivative operators are very useful in the comprehension of the structure of the M_L spaces. We describe here how a Schur function partial derivative operator acts on lattice diagrams with distinct cells in the positive quadrant. | Schur Partial Derivative Operators | 13,639 |
The domatic number of a graph $G$, denoted $dom(G)$, is the maximum possible cardinality of a family of disjoint sets of vertices of $G$, each set being a dominating set of $G$. It is well known that every graph without isolated vertices has $dom(G) \geq 2$. For every $k$, it is known that there are graphs with minimum degree at least $k$ and with $dom(G)=2$. In this paper we prove that this is not the case if $G$ is $k$-regular or {\em almost} $k$-regular (by ``almost'' we mean that the minimum degree is $k$ and the maximum degree is at most $Ck$ for some fixed real number $C \geq 1$). In this case we prove that $dom(G) \geq (1+o_k(1))k/(2\ln k)$. We also prove that the order of magnitude $k/\ln k$ cannot be improved. One cannot replace the constant 2 with a constant smaller than 1. The proof uses the so called {\em semi-random method} which means that combinatorial objects are generated via repeated applications of the probabilistic method; in our case iterative applications of the Lov\'asz Local Lemma. | The domatic number of regular and almost regular graphs | 13,640 |
We verify the Upper Bound Conjecture (UBC) for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes -- a short simplicial h-vector. | A short simplicial h-vector and the upper bound theorem | 13,641 |
In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its description as the intersection of halfspaces, which determine the facets of the polytope. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We proved that, if our polytope is a simplex, the lattice point counts in the interior and closure of such a vector-dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes. In the present paper we complete the picture by extending this result to general rational polytopes. As a corollary, we also generalize a reciprocity theorem of Stanley. | Multidimensional Ehrhart Reciprocity | 13,642 |
Following Mansour, let $S_n^{(r)}$ be the set of all coloured permutations on the symbols $1,2,...,n$ with colours $1,2,...,r$, which is the analogous of the symmetric group when r=1, and the hyperoctahedral group when r=2. Let $I\subseteq\{1,2,...,r\}$ be subset of d colours; we define $T_{k,r}^m(I)$ be the set of all coloured permutations $\phi\in S_k^{(r)}$ such that $\phi_1=m^{(c)}$ where $c\in I$. We prove that, the number $T_{k,r}^m(I)$-avoiding coloured permutations in $S_n^{(r)}$ equals $(k-1)!r^{k-1}\prod_{j=k}^n h_j$ for $n\geq k$ where $h_j=(r-d)j+(k-1)d$. We then prove that for any $\phi\in T_{k,r}^1(I)$ (or any $\phi\in T_{k,r}^k(I)$), the number of coloured permutations in $S_n^{(r)}$ which avoid all patterns in $T_{k,r}^1(I)$ (or in $T_{k,r}^k(I)$) except for $\phi$ and contain $\phi$ exactly once equals $\prod_{j=k}^n h_j\cdot \sum_{j=k}^n \frac{1}{h_j}$ for $n\geq k$. Finally, for any $\phi\in T_{k,r}^m(I)$, $2\leq m\leq k-1$, this number equals $\prod_{j=k+1}^n h_j$ for $n\geq k+1$. These results generalize recent results due to Mansour, and due to Simion. | Coloured permutations containing and avoiding certain patterns | 13,643 |
Sperner's bound on the size of an antichain in the lattice P(S) of subsets of a finite set S has been generalized in three different directions: by Erdos to subsets of P(S) in which chains contain at most r elements; by Meshalkin to certain classes of compositions of S; by Griggs, Stahl, and Trotter through replacing the antichains by certain sets of pairs of disjoint elements of P(S). We unify Erdos's, Meshalkin's, and Griggs-Stahl-Trotter's inequalities with a common generalization. We similarly unify their accompanying LYM inequalities. Our bounds do not in general appear to be the best possible. | A unifying generalization of Sperner's theorem | 13,644 |
Meshalkin's theorem states that a class of ordered p-partitions of an n-set has at most $\max \binom{n}{a_1,...,a_p}$ members if for each k the k'th parts form an antichain. We give a new proof of this and the corresponding LYM inequality due to Hochberg and Hirsch, which is simpler and more general than previous proofs. It extends to a common generalization of Meshalkin's theorem and Erdos's theorem about r-chain-free set families. | A shorter, simpler, stronger proof of the Meshalkin-Hochberg-Hirsch
bounds on componentwise antichains | 13,645 |
Let M be a family of sequences (a_1,...,a_p) where each a_k is a flat in a projective geometry of rank n (dimension n-1) and order q, and the sum of ranks, r(a_1) + ... + r(a_p), equals the rank of the join a_1 v ... v a_p. We prove upper bounds on |M| and corresponding LYM inequalities assuming that (i) all joins are the whole geometry and for each k<p the set of all a_k's of sequences in M contains no chain of length l, and that (ii) the joins are arbitrary and the chain condition holds for all k. These results are q-analogs of generalizations of Meshalkin's and Erdos's generalizations of Sperner's theorem and their LYM companions, and they generalize Rota and Harper's q-analog of Erdos's generalization. | A Meshalkin theorem for projective geometries | 13,646 |
We consider the problem of enumerating the permutations containing exactly $k$ occurrences of a pattern of length 3. This enumeration has received a lot of interest recently, and there are a lot of known results. This paper presents an alternative approach to the problem, which yields a proof for a formula which so far only was conjectured (by Noonan and Zeilberger). This approach is based on bijections from permutations to certain lattice paths with ``jumps'', which were first considered by Krattenthaler. | Enumeration of permutations containing a prescribed number of
occurrences of a pattern of length 3 | 13,647 |
We present a bijective proof of the hook-length formula for shifted standard tableaux of a fixed shape based on a modified jeu de taquin and the ideas of the bijective proof of the hook-length formula for ordinary standard tableaux by Novelli, Pak and Stoyanovskii. In their proof Novelli, Pak and Stoyanovskii define a bijection between arbitrary fillings of the Ferrers diagram with the integers $1,2,...,n$ and pairs of standard tableaux and hook tabloids. In our shifted version of their algorithm the map from the set of arbitrary fillings of the shifted Ferrers diagram onto the set of shifted standard tableaux is analog to the construction of Novelli, Pak and Stoyanovskii, however, unlike to their algorithm, we are forced to use the 'rowwise' total order of the cells in the shifted Ferrers diagram rather than the 'columnwise' total order as the underlying order in the algorithm. Unfortunately the construction of the shifted hook tabloid is more complicated in the shifted case. As a side-result we obtain a simple random algorithm for generating shifted standard tableaux of a given shape, which produces every such tableau equally likely. | A bijective proof of the hook-length formula for shifted standard
tableaux | 13,648 |
For their bijective proof of the hook-length formula for the number of standard tableaux of a fixed shape Novelli, Pak and Stoyanovskii define a modified jeu de taquin which transforms an arbitrary filling of the Ferrers diagram with $1,2,...,n$ (tabloid) into a standard tableau. Their definition relies on a total order of the cells in the Ferrers diagram induced by a special standard tableau, however, this definition also makes sense for the total order induced by any other standard tableau. Given two standard tableaux $P,Q$ of the same shape we show that the number of tabloids which result in $P$ if we perform modified jeu de taquin with respect to the total order induced by $Q$ is equal to the number of tabloids which result in $Q$ if we perform modified jeu de taquin with respect to $P$. This symmetry theorem extends to skew shapes and shifted skew shapes. | A symmetry theorem on a modified jeu de taquin | 13,649 |
This paper discusses reformulations of the problem of coloring plane maps with four colors. The context is the edge-coloring with three colors of cubic graphs such that three distinct colors occur at each vertex. We include discussion of the Eliahou-Kryuchkov conjecture, the Penrose formula, the vector cross product formulation and the reformulations in terms of formations and factorizations due to G. Spencer-Brown. The latter includes a proof of the Spencer-Brown parity lemma and discussion of the parity-pass algorithm. | Reformulating the Map Color Theorem | 13,650 |
We study a certain poset on the free monoid on a countable alphabet. This poset is determined by the fact that its total extensions are precisely the standard term orders. We also investigate the poset classifying degree-compatible standard term orders, and the poset classifying sorted term orders. For the latter poset, we give a Galois coconnection with the Young lattice. | A poset classifying non-commutative term orders | 13,651 |
We study generating functions for the number of involutions in $S_n$ avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation $\tau$ on $k$ letters. In several interesting cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both 132 and $12... k$ have the same enumerative formula according to the length than involutions avoiding both 132 and any {\em double-wedge pattern} possibly followed by fixed points of total length $k$. Many results are also shown with a combinatorial point of view. | Restricted 132-Involutions and Chebyshev Polynomials | 13,652 |
Given an m x n rectangular mesh, its adjacency matrix A, having only integer entries, may be interpreted as a map between vector spaces over an arbitrary field K. We describe the kernel of A: it is a direct sum of two natural subspaces whose dimensions are equal to $\lceil c/2 \rceil$ and $\lfloor c/2 \rfloor$, where c = gcd (m+1,n+1) - 1. We show that there are bases to both vector spaces, with entries equal to 0, 1 and -1. When K = Z/(2), the kernel elements of these subspaces are described by rectangular tilings of a special kind. As a corollary, we count the number of tilings of a rectangle of integer sides with a specified set of tiles. | The kernel of the adjacency matrix of a rectangular mesh | 13,653 |
Actual individual preferences are neither complete (=total) nor antisymmetric in general, so that at least every quasi-order must be an admissible input to a satisfactory choice rule. It is argued that the traditional notion of ``indifference'' in individual preferences is misleading and should be replaced by `equivalence' and `undecidedness'. In this context, ten types of majority and minority arguments of different strength are studied which lead to social choice rules that accept profiles of arbitrary reflexive relations. These rules are discussed by means of many familiar, and some new conditions, including `immunity from binary arguments'. Moreover, it is proved that every choice function satisfying two weak Condorcet-type conditions can be made both composition-consistent and idempotent, and that all the proposed rules have polynomial time complexity. | Social Choice Under Incomplete, Cyclic Preferences | 13,654 |
In this paper, we find an explicit formulas, or recurrences, in terms of generating functions for the cardinalities of the sets $S_n(T;\tau)$ of all permutations in $S_n$ that contain $\tau\in S_k$ exactly once and avoid a subset $T\subseteq S_3$, $|T|\geq2$. The main body of the paper is divided into three sections corresponding to the cases $|T|=2,3$ and $|T|\geq4$. | Permutations containing a pattern exactly once and avoiding at least two
patterns of three letters | 13,655 |
We present new functional equations for the species of plane and of planar (in the sense of Harary and Palmer, 1973) 2-trees and some associated pointed species. We then deduce the explicit molecular expansion of these species, i.e a classification of their structures according to their stabilizers. There result explicit formulas in terms of Catalan numbers for their associated generating series, including the asymmetry index series. This work is closely related to the enumeration of polyene hydrocarbons of molecular formula C_nH_n+2. | A classification of plane and planar 2-trees | 13,656 |
The aim of this work is to study the quotient ring R_n of the ring Q[x_1,...,x_n] over the ideal J_n generated by non-constant homogeneous quasi-symmetric functions. We prove here that the dimension of R_n is given by C_n, the n-th Catalan number. This is also the dimension of the space SH_n of super-covariant polynomials, that is defined as the orthogonal complement of J_n with respect to a given scalar product. We construct a basis for R_n whose elements are naturally indexed by Dyck paths. This allows us to understand the Hilbert series of SH_n in terms of number of Dyck paths with a given number of factors. | Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for
S_n | 13,657 |
Borsuk's conjecture states that any bounded set in R^n can be partitioned into n+1 sets of smaller diameter. It is known to be false for all n bigger or equal to 323. Here we show that Borsuk's conjecture fails in dimensions 321 and 322. (This result has been independently discovered by Hinrichs and Richter.) | Borsuk's Conjecture Fails in Dimensions 321 and 322 | 13,658 |
Locally finite self-similar graphs with bounded geometry and without bounded geometry as well as non-locally finite self-similar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The length scaling factor $\nu$ and the volume scaling factor $\mu$ can be defined similarly to the corresponding parameters of continuous self-similar sets. There are different notions of growth dimensions of graphs. For a rather general class of self-similar graphs it is proved that all these dimensions coincide and that they can be calculated in the same way as the Hausdorff dimension of continuous self-similar fractals: \[\dim X=\frac{\log \mu}{\log \nu}.\] | Growth of self-similar graphs | 13,659 |
A natural generalization of single pattern avoidance is subset avoidance. A complete study of subset avoidance for the case k=3 is carried out in [SS]. For k>3 situation becomes more complicated, as the number of possible cases grows rapidly. Recently, several authors have considered the case of general k when T has some nice algebraic properties. Barcucci, Del Lungo, Pergola, and Pinzani in [BDPP) treated the case when $T=T_1$ is the centralizer of k-1 and k under the natural action of $S_k$ on [k]. Mansour and Vainshtein in [MVp] treated the case when $T=T_2$ is maximal parabolic group of $S_k$. Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In this paper we present an analogue with generalization for the case $T_1$ and for the case $T_2$ by using generalized patterns instead of classical patterns. | The centralizer of two numbers under the natural action of $S_k$ on [k],
the maximal parabolic subgroup of $S_k$, and generalized patterns | 13,660 |
Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In this paper we study the generating functions for the number of permutations on $n$ letters avoiding a generalized pattern $ab\mn c$ where $(a,b,c)\in S_3$, and containing a prescribed number of occurrences of generalized pattern $cd\mn e$ where $(c,d,e)\in S_3$. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results. | Restricted permutations by patterns of type $(2,1)$ | 13,661 |
Let $H$ be a $k$-uniform hypergraph with $n$ vertices. A {\em strong $r$-coloring} is a partition of the vertices into $r$ parts, such that each edge of $H$ intersects each part. A strong $r$-coloring is called {\em equitable} if the size of each part is $\lceil n/r \rceil$ or $\lfloor n/r \rfloor$. We prove that for all $a \geq 1$, if the maximum degree of $H$ satisfies $\Delta(H) \leq k^a$ then $H$ has an equitable coloring with $\frac{k}{a \ln k}(1-o_k(1))$ parts. In particular, every $k$-uniform hypergraph with maximum degree $O(k)$ has an equitable coloring with $\frac{k}{\ln k}(1-o_k(1))$ parts. The result is asymptotically tight. The proof uses a double application of the non-symmetric version of the Lov\'asz Local Lemma. | Equitable coloring of k-uniform hypergraphs | 13,662 |
Let $H$ be a hypergraph. For a $k$-edge coloring $c : E(H) \to \{1,...,k\}$ let $f(H,c)$ be the number of components in the subhypergraph induced by the color class with the least number of components. Let $f_k(H)$ be the maximum possible value of $f(H,c)$ ranging over all $k$-edge colorings of $H$. If $H$ is the complete graph $K_n$ then, trivially, $f_1(K_n)=f_2(K_n)=1$. In this paper we prove that for $n \geq 6$, $f_3(K_n)=\lfloor n/6 \rfloor+1$ and supply close upper and lower bounds for $f_k(K_n)$ in case $k \geq 4$. Several results concerning the value of $f_k(K_n^r)$, where $K_n^r$ is the complete $r$-uniform hypergraph on $n$ vertices, are also established. | Edge coloring complete uniform hypergraphs with many components | 13,663 |
Given a finite set of vectors spanning a lattice and lying in a halfspace of a real vector space, to each vector $a$ in this vector space one can associate a polytope consisting of nonnegative linear combinations of the vectors in the set which sum up to $a$. This polytope is called the partition polytope of $a$. If $a$ is integral, this polytope contains a finite set of lattice points corresponding to nonnegative integral linear combinations. The partition polytope associated to an integral $a$ is a rational convex polytope, and any rational convex polytope can be realized canonically as a partition polytope. We consider the problem of counting the number of lattice points in partition polytopes, or, more generally, computing sums of values of exponential-polynomial functions on the lattice points in such polytopes. We give explicit formulae for these quantities using a notion of multi-dimensional residue due to Jeffrey-Kirwan. We show, in particular, that the dependence of these quantities on $a$ is exponential-polynomial on "large neighborhoods" of chambers. Our method relies on a theorem of separation of variables for the generating function, or, more generally, for periodic meromorphic functions with poles on an arrangement of affine hyperplanes. | Residue formulae for vector partitions and Euler-MacLaurin sums | 13,664 |
The n'th Birkhoff polytope is the set of all doubly stochastic n-by-n matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A wide open problem concerns the volumes of these polytopes, which have been known for n up to 8. We present a new, complex-analytic way to compute the Ehrhart polynomial of the Birkhoff polytope, that is, the function counting the integer points in the dilated polytope. One reason to be interested in this counting function is that the leading term of the Ehrhart polynomial is--up to a trivial factor--the volume of the polytope. We implemented our methods in form of a computer program, which yielded the Ehrhart polynomial (and hence the volume) of the ninth Birkhoff polytope. | The Ehrhart polynomial of the Birkhoff polytope | 13,665 |
We describe Maple packages for the automatic generation of generating functions(and series expansions) for counting lattice animals(fixed polyominoes), in the two-dimensional hexagonal lattice, of bounded but arbitrary width. Our Maple packages(complete with source code) are easy-to-use and available from my website. | Counting Hexagonal Lattice Animals | 13,666 |
Define $S_n^k(\alpha)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_m$. Let $s_n^k(\alpha)$ be the size of $S_n^k(\alpha)$. We investigate $S_n^0(\alpha)$ for all $\alpha \in S_3$ as well as show that $s_n^k(132)=s_n^k(213)=s_n^k(321)$ and $s_n^k(231)=s_n^k(312)$ for all $0 \leq k \leq n$. | Refined Restricted Permutations | 13,667 |
In this paper we present a novel project-and-lift approach to compute the set of minimal generators of the semigroup $(\Lambda\cap\R^n_+,+)$ for lattices $\Lambda\subseteq\Z^n$. This problem class includes the computation of Hilbert bases of cones $\{z:Az=0,z\in\R^n_+\}$ for integer matrices $A$. A similar approach can be used to compute only the extreme rays of such cones. Finally, some combinatorial applications and computational experience are presented. | On the Computation of Hilbert Bases and Extreme Rays of Cones | 13,668 |
A Dyck path is a lattice path in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of steps (1,1) and (1,-1), which never passes below the x-axis. A peak at height k on a Dyck path is a point on the path with coordinate y=k that is immediately preceded by a (1,1) step and immediately followed by a (1,-1) step. In this paper we find an explicit expression to the generating function for the number of Dyck paths starting at (0,0) and ending at (2n,0) with exactly r peaks at height k. This allows us to express this function via Chebyshev polynomials of the second kind and generating function for the Catalan numbers. | Counting peaks at height k in a Dyck path | 13,669 |
A permutation $\pi \in S_n$ is said to {\it avoid} a permutation $\sigma \in S_k$ whenever $\pi$ contains no subsequence with all of the same pairwise comparisons as $\sigma$. For any set $R$ of permutations, we write $S_n(R)$ to denote the set of permutations in $S_n$ which avoid every permutation in $R$. In 1985 Simion and Schmidt showed that $|S_n(132, 213, 123)|$ is equal to the Fibonacci number $F_{n+1}$. In this paper we generalize this result in several ways. We first use a result of Mansour to show that for any permutation $\tau$ in a certain infinite family of permutations, $|S_n(132, 213, \tau)|$ is given in terms of Fibonacci numbers or $k$-generalized Fibonacci numbers. In many cases we give explicit enumerations, which we prove bijectively. We then use generating function techniques to show that for any permutation $\gamma$ in a second infinite family of permutations, $|S_n(123, 132, \gamma)|$ is also given in terms of Fibonacci numbers or $k$-generalized Fibonacci numbers. In many cases we give explicit enumerations, some of which we prove bijectively. We go on to use generating function techniques to show that for any permutation $\omega$ in a third infinite family of permutations, $|S_n(132, 2341, \omega)|$ is given in terms of Fibonacci numbers, and for any permutation $\mu$ in a fourth infinite family of permutations, $|S_n(132, 3241, \mu)|$ is given in terms of Fibonacci numbers and $k$-generalized Fibonacci numbers. In several cases we give explicit enumerations. We conclude by giving an infinite class of examples of a set $R$ of permutations for which $|S_n(R)|$ satisfies a linear homogeneous recurrence relation with constant coefficients. | Restricted Permutations, Fibonacci Numbers, and k-generalized Fibonacci
Numbers | 13,670 |
We show that a graph has an orientation under which every circuit of even length is clockwise odd if and only if the graph contains no subgraph which is, after the contraction of at most one circuit of odd length, an even subdivision of K_{2,3}. In fact we give a more general characterisation of graphs that have an orientation under which every even circuit has a prescribed clockwise parity. This problem was motivated by the study of Pfaffian graphs, which are the graphs that have an orientation under which every alternating circuit is clockwise odd. Their significance is that they are precisely the graphs to which Kasteleyn's powerful method for enumerating perfect matchings may be applied. | Even circuits of prescribed clockwise parity | 13,671 |
Schanuel has pointed out that there are mathematically interesting categories whose relationship to the ring of integers is analogous to the relationship between the category of finite sets and the semi-ring of non-negative integers. Such categories are inherently geometrical or topological, in that the mapping to the ring of integers is a variant of Euler characteristic. In these notes, I sketch some ideas that might be used in further development of a theory along lines suggested by Schanuel. | Euler measure as generalized cardinality | 13,672 |
Two of the pillars of combinatorics are the notion of choosing an arbitrary subset of a set with $n$ elements (which can be done in $2^n$ ways), and the notion of choosing a $k$-element subset of a set with $n$ elements (which can be done in $n \choose k$ ways). In this article I sketch the beginnings of a theory that would import these notions into the category of hedral sets in the sense of Morelli and the category of polyhedral sets in the sense of Schanuel. Both of these theories can be viewed as extensions of the theory of finite sets and mappings between finite sets, with the concept of cardinality being replaced by the more general notion of Euler measure (sometimes called combinatorial Euler characteristic). I prove a ``functoriality'' theorem (Theorem 1) for subset-selection in the context of polyhedral sets, which provides quasi-combinatorial interpretations of assertions such as $2^{-1} = \frac12$ and ${1/2 \choose 2} = -\frac18$. Furthermore, the operation of forming a power set can be viewed as a special case of the operation of forming the set of all mappings from one set to another; I conclude the article with a polyhedral analogue of the set of all mappings between two finite sets, and a restrictive but suggestive result (Theorem 2) that offers a hint of what a general theory of exponentiation in the polyhedral category might look like. | Exponentiation and Euler measure | 13,673 |
Chen's lemma on iterated integrals implies that certain identities involving multiple integrals, such as the de Bruijn and Wick formulas, amount to combinatorial identities for Pfaffians and hafnians in shuffle algebras. We provide direct algebraic proofs of such shuffle identities, and obtain various generalizations. We also discuss some Pfaffian identities due to Sundquist and Ishikawa-Wakayama, and a Cauchy formula for anticommutative symmetric functions. Finally, we extend some of the previous considerations to hyperpfaffians and hyperhafnians. | Pfaffian and hafnian identities in shuffle algebras | 13,674 |
Let $P$ and $Q$ be polytopes, the first of "low" dimension and the second of "high" dimension. We show how to triangulate the product $P \times Q$ efficiently (i.e., with few simplices) starting with a given triangulation of $Q$. Our method has a computational part, where we need to compute an efficient triangulation of $P \times \Delta^m$, for a (small) natural number $m$ of our choice. $\Delta^m$ denotes the $m$-simplex. Our procedure can be applied to obtain (asymptotically) efficient triangulations of the cube $I^n$: We decompose $I^n = I^k \times I^{n-k}$, for a small $k$. Then we recursively assume we have obtained an efficient triangulation of the second factor and use our method to triangulate the product. The outcome is that using $k=3$ and $m=2$, we can triangulate $I^n$ with $O(0.816^{n} n!)$ simplices, instead of the $O(0.840^{n} n!)$ achievable before. | Asymptotically efficient triangulations of the d-cube | 13,675 |
A recently posed question of Haggkvist and Scott's asked whether or not there exists a constant c such that if G is a graph of minimum degree ck then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that for k > 1, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2-1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k-1)n^{1 + 1/k} has a cycle of length 2k. | Arithmetic Progressions of Cycle Lengths in Graphs | 13,676 |
We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3-letter subword pattern. | Counting occurrences of some subword patterns | 13,677 |
In [BabStein] Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper we find the exponential generating function for the remaining case. Also we consider permutations that avoid a pattern of the form $x-yz$ or $xy-z$ and begin with one of the patterns $12... k$, $k(k-1)... 1$, $23... k1$, $(k-1)(k-2)... 1k$ or end with one of the patterns $12... k$, $k(k-1)... 1$, $1k(k-1)... 2$, $k12... (k-1)$. For each of these cases we find either the ordinary or exponential generating functions or a precise formula for the number of such permutations. Besides we generalize some of the obtained results as well as some of the results given in [Kit3]: we consider permutations avoiding certain generalized 3-patterns and beginning (ending) with an arbitrary pattern having either the greatest or the least letter as its rightmost (leftmost) letter. | Simultaneous avoidance of generalized patterns | 13,678 |
In 1990 West conjectured that there are $2(3n)!/((n+1)!(2n+1)!)$ two-stack sortable permutations on $n$ letters. This conjecture was proved analytically by Zeilberger in 1992. Later, Dulucq, Gire, and Guibert gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on $n$ letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation $\tau$ on $k$ letters. In several interesting cases this generating function can be expressed in terms of the generating function for the Fibonacci numbers or the generating function for the Pell numbers. | 132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and
Pell Numbers | 13,679 |
Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider n-permutations that avoid the generalized pattern 1-32 and whose k rightmost letters form an increasing subword. The number of such permutations is a linear combination of Bell numbers. We find a bijection between these permutations and all partitions of an $(n-1)$-element set with one subset marked that satisfy certain additional conditions. Also we find the e.g.f. for the number of permutations that avoid a generalized 3-pattern with no dashes and whose k leftmost or k rightmost letters form either an increasing or decreasing subword. Moreover, we find a bijection between n-permutations that avoid the pattern 132 and begin with the pattern 12 and increasing rooted trimmed trees with n+1 nodes. | Generalized pattern avoidance with additional restrictions | 13,680 |
Berstel proved that the Arshon sequence cannot be obtained by iteration of a morphism. An alternative proof of this fact is given here. The $\sigma$-sequence was constructed by Evdokimov in order to construct chains of maximal length in the n-dimensional unit cube. It turns out that the $\sigma$-sequence has a close connection to the Dragon curve. We prove that the $\sigma$-sequence can not be defined by iteration of a morphism. | There are no iterated morphisms that define the Arshon sequence and the
$σ$-sequence | 13,681 |
We introduced the notation of a set of prohibitions and give definitions of a complete set and a crucial word with respect to a given set of prohibitions. We consider 3 particular sets which appear in different areas of mathematics and for each of them examine the length of a crucial word. One of these sets is proved to be incomplete. The problem of determining lengths of words that are free from a set of prohibitions is shown to be NP-complete, although the related problem of whether or not a given set of prohibitions is complete is known to be effectively solvable. | Crucial Words and the Complexity of Some Extremal Problems for Sets of
Prohibited Words | 13,682 |
This paper studies the problem of reconstructing binary matrices that are only accessible through few evaluations of their discrete X-rays. Such question is prominently motivated by the demand in material science for developing a tool for the reconstruction of crystalline structures from their images obtained by high-resolution transmission electron microscopy. Various approaches have been suggested for solving the general problem of reconstructing binary matrices that are given by their discrete X-rays in a number of directions, but more work have to be done to handle the ill-posedness of the problem. We can tackle this ill-posedness by limiting the set of possible solutions, by using appropriate a priori information, to only those which are reasonably typical of the class of matrices which contains the unknown matrix that we wish to reconstruct. Mathematically, this information is modelled in terms of a class of binary matrices to which the solution must belong. Several papers study the problem on classes of binary matrices on which some connectivity and convexity constraints are imposed. We study the reconstruction problem on some new classes consisting of binary matrices with periodicity properties, and we propose a polynomial-time algorithm for reconstructing these binary matrices from their orthogonal discrete X-rays. | Discrete Tomography: Reconstruction under periodicity constraints | 13,683 |
Given an rc-graph $R$ of permutation $w$ and an rc-graph $Y$ of permutation $v$, we provide an insertion algorithm, which defines an rc-graph $R\leftarrow Y$ in the case when $v$ is a shuffle with the descent at $r$ and $w$ has no descents greater than $r$ or in the case when $v$ is a shuffle, whose shape is a hook. This algorithm gives a combinatorial rule for computing the generalized Littlewood-Richardson coefficients $c^{u}_{wv}$ in the two cases mentioned above. | Generalization of Schensted insertion algorithm to the cases of hooks
and semi-shuffles | 13,684 |
In this paper we consider a variation of the classical Tur\'{a}n-type extremal problems. Let $S$ be an $n$-term graphical sequence, and $\sigma(S)$ be the sum of the terms in $S$. Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$-term graphical sequence $S$ having $\sigma(S)\ge l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $\sigma(H, n)$. We show $\sigma(C_{2m+1}, n)=m(2n-m-1)+2$, for $m\ge 3$, $n\ge 3m$; $\sigma(C_{2m+2}, n)=m(2n-m-1)+4$, for $m\ge 3, n\ge 5m-2$. | The smallest degree sum that yields potentially $C_k$-graphical sequence | 13,685 |
In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+32t-1$$ for $t=27720r+169 (r\geq 1)$ and $n\geq{6911/16}t^{2}+{514441/8}t-{3309665/16}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + {2562 \over 6911}}.$ | A Lower Bound for the Number of Edges in a Graph Containing No Two
Cycles of the Same Length | 13,686 |
In this article, we discuss some classical problems in combinatorics which can be solved by exploiting analogues between graph theory and the theory of manifolds. One well-known example is the McMullen conjecture, which was settled twenty years ago by Richard Stanley by interpreting certain combinatorial invariants of convex polytopes as the Betti numbers of a complex projective variety. Another example is the classical parallel redrawing problem, which turns out to be closely related to the problem of computing the second Betti number of a complex compact $(\C^*)^n$-manifold. | How is a graph like a manifold? | 13,687 |
Let w_0 denote the permutation [n,n-1,...,2,1]. We give two new explicit formulae for the Kazhdan-Lusztig polynomials P_{w_0w,w_0x} in S_n when x is a maximal element in the singular locus of the Schubert variety X_w. To do this, we utilize a standard identity that relates P_{x,w} and P_{w_0w,w_0x}. | Two formulae for inverse Kazhdan-Lusztig polynomials in S_n | 13,688 |
In [GM] Guibert and Mansour studied involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern on k letters. They also established a bijection between 132-avoiding involutions and Dyck word prefixes of same length. Extending this bijection to bilateral words allows to determine more parameters; in particular, we consider the number of inversions and rises of the involutions onto the words. This is the starting point for considering two different directions: even/odd involutions and statistics of some generalized patterns. Thus we first study generating functions for the number of even or odd involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern $\tau$ on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. Next, we consider other statistics on 132-avoiding involutions by counting an occurrences of some generalized patterns, related to the enumeration according to the number of rises. | Some statistics on restricted 132 involutions | 13,689 |
The k-th power D^k of a directed graph D is defined to be the directed graph on the vertices of D with an arc from a to b in D^k iff one can get from a to b in D with exactly k steps. This notion is equivalent to the k-fold composition of binary relations or k-th powers of Boolean matrices. A k-th root of a directed graph D is another directed graph R with R^k = D. We show that for each k >= 2, computing a k-th root of a directed graph is at least as hard as the graph isomorphism problem. | Computing roots of directed graphs is graph isomorphism hard | 13,690 |
A construction for sphere packings is introduced that is parallel to the ``anticode'' construction for codes. This provides a simple way to view Vardy's recent 20-dimensional sphere packing, and also produces packings in dimensions 22, 44--47 that are denser than any previously known. | The Antipode Construction for Sphere Packings | 13,691 |
We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, along with a general design procedure. This procedure is illustrated using the Z^2 lattice. The asymptotic performance of the quantizer is analyzed in the high-rate case. We also evaluate its rate-distortion performance and compare it to known information theoretic bounds. | Asymmetric Multiple Description Lattice Vector Quantizers | 13,692 |
Subset take-away is a two-player game involving a fixed finite set A. Players alternate choosing a proper, non-empty subset of A, with the condition that one may not name a set containing a set that was named earlier. A player unable to move loses. It was conjectured by David Gale that this game is always a second player win, and this was known to hold if A has no more than 5 elements. In this paper, we describe a technique called "binary star reduction" that often allows one to dramatically reduce the complexity of a position. Using this tool and some computer search we show that Gale's conjecture holds when A has six elements. We also show how this game can be interpreted geometrically. | David Gale's subset take-away game | 13,693 |
A 321-k-gon-avoiding permutation pi avoids 321 and the following four patterns: k(k+2)(k+3)...(2k-1)1(2k)23...(k+1), k(k+2)(k+3)...(2k-1)(2k)123...(k+1), (k+1)(k+2)(k+3)...(2k-1)1(2k)23...k, (k+1)(k+2)(k+3)...(2k-1)(2k)123...k. The 321-4-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincare polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW] gave an exact enumeration in terms of linear recurrences with constant coefficients for the cases k=2,3,4. In this paper, we extend these results by finding an explicit expression for the generating function for the number of 321-k-gon-avoiding permutations on n letters. The generating function is expressed via Chebyshev polynomials of the second kind. | 321-polygon-avoiding permutations and Chebyshev polynomials | 13,694 |
Let B_n be the hyperoctahedral group; that is, the set of all signed permutations on n letters, and let B_n(T) be the set of all signed permutations in B_n which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets B_n(T) where $T \subseteq B_2$. This allow us to express these cardinalities via inverse of binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers. | Avoiding 2-letter signed patterns | 13,695 |
Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N=24, 26, >= 28; 7-designs for N=24, 30, 32, 34, >= 36; 8-designs for N=36, 40, 42, >= 44; 9-designs for N=48, 50, 52, >= 54; 10-designs for N=60, 62, >= 64; 11-designs for N=70, 72, >= 74; and 12-designs for N=84, >= 86. The existence of some of these designs is established analytically, while others are given by very accurate numerical coordinates. The 24-point 7-design was first found by McLaren in 1963, and -- although not identified as such by McLaren -- consists of the vertices of an "improved" snub cube, obtained from Archimedes' regular snub cube (which is only a 3-design) by slightly shrinking each square face and expanding each triangular face. 5-designs with 23 and 25 points are presented which, taken together with earlier work of Reznick, show that 5-designs exist for N=12, 16, 18, 20, >= 22. It is conjectured, albeit with decreasing confidence for t >= 9, that these lists of t-designs are complete and that no others exist. One of the constructions gives a sequence of putative spherical t-designs with N= 12m points (m >= 2) where N = t^2/2 (1+o(1)) as t -> infinity. | McLaren's Improved Snub Cube and Other New Spherical Designs in Three
Dimensions | 13,696 |
The problem of designing a multiple description vector quantizer with lattice codebook Lambda is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance results are obtained for quantizers based on the lattices A_2 and Z^i, i=1,2,4,8, that make use of this labeling algorithm. The high-rate squared-error distortions for this family of L-dimensional vector quantizers are then analyzed for a memoryless source with probability density function p and differential entropy h(p) < infty. For any a in (0,1) and rate pair (R,R), it is shown that the two-channel distortion d_0 and the channel 1 (or channel 2) distortions d_s satisfy lim_{R -> infty} d_0 2^(2R(1+a)) = (1/4) G(Lambda) 2^{2h(p)} and lim_{R -> infty} d_s 2^(2R(1-a)) = G(S_L) 2^2h(p), where G(Lambda) is the normalized second moment of a Voronoi cell of the lattice Lambda and G(S_L) is the normalized second moment of a sphere in L dimensions. | Multiple Description Vector Quantization with Lattice Codebooks: Design
and Analysis | 13,697 |
The highest possible minimal norm of a unimodular lattice is determined in dimensions n <= 33. There are precisely five odd 32-dimensional lattices with the highest possible minimal norm (compared with more than 8*10^20 in dimension 33). Unimodular lattices with no roots exist if and only if n >= 23, n not = 25. | A Note on Optimal Unimodular Lattices | 13,698 |
It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[n/24] +2, unless n = 23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the bound for even unimodular lattices to strongly N-modular even lattices for N in {1,2,3,5,6,7,11,14,15,23} ... (*), and analogous bounds are established here for odd lattices satisfying certain technical conditions (which are trivial for N = 1 and 2). For N > 1 in (*), lattices meeting the new bound are constructed that are analogous to the ``shorter'' and ``odd'' Leech lattices. These include an odd associate of the 16-dimensional Barnes-Wall lattice and shorter and odd associates of the Coxeter-Todd lattice. A uniform construction is given for the (even) analogues of the Leech lattice, inspired by the fact that (*) is also the set of square-free orders of elements of the Mathieu group M_{23}. | The Shadow Theory of Modular and Unimodular Lattices | 13,699 |
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