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The convolved Fibonacci numbers F_j^(r) are defined by (1-z-z^2)^{-r}=\sum_{j>=0}F_{j+1}^(r)z^j. In this note some related numbers that can be expressed in terms of convolved Fibonacci numbers are considered. These numbers appear in the numerical evaluation of a certain number theoretical constant. This note is a case study of the transform {1/n}\sum_{d|n}mu(d)f(z^d)^{n/d}, with f any formal series and mu the Moebius function), which is studied in a companion paper entitled `The formal series Witt transform'. | Convoluted convolved Fibonacci numbers | 13,900 |
We study the distribution of the statistics 'number of fixed points' and 'number of excedances' in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions enumerating these two statistics. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statistics in involutions avoiding any subset of patterns of length 3. The main technique is to use bijections between pattern-avoiding permutations and certain kinds of Dyck paths, in such a way that the statistics in permutations that we study correspond to statistics on Dyck paths that are easy to enumerate. | Multiple pattern avoidance with respect to fixed points and excedances | 13,901 |
An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M-partition is a log-linear function of m and the M-partitions of m correspond to lattice points in a polytope. We exhibit a recurrence relation for counting the number of M-partitions of m and, for ``half'' of the positive integers, this recurrence relation will have a generating function. The generating function will be, in some sense, the same as the generating function for counting the number of distinct binary partitions for a given integer. | M-partitions: Optimal partitions of weight for one scale pan | 13,902 |
We present a complete solution to the so-called tennis ball problem, which is equivalent to counting lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit expressions for the corresponding generating functions. Our method is based on the properties of Tutte polynomials of matroids associated to lattice paths. We also show how the same method provides a solution to a wide generalization of the problem. | A solution to the tennis ball problem | 13,903 |
Lajos Takacs gave a somewhat formidable alternating sum formula for the number of forests of unrooted trees on $n$ labeled vertices. Here we use a weight-reversing involution on suitable tree configurations to give a combinatorial derivation of Takacs' formula. | A combinatorial derivation of the number of labeled forests | 13,904 |
We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the $CL$-shellability criterion of Bj\"orner and Wachs for posets and its generalization by Kozlov called $CC$-shellability. We give a lexicographic shelling for the quotient of the order complex of a Boolean algebra of rank $2n$ by the action of the wreath product $S_2\wr S_n$ of symmetric groups, and we provide a partitioning for the quotient complex $\Delta (\Pi_n)/S_n $. Stanley asked for a description of the symmetric group representation $\beta_S $ on the homology of the rank-selected partition lattice $\Pi_n^S $ in [St2], and in particular he asked when the multiplicity $b_S(n)$ of the trivial representation in $\beta_S$ is 0. One consequence of the partitioning for $\dps $ is a (fairly complicated) combinatorial interpretation for $b_S(n) $; another is a simple proof of Hanlon's result that $b_{1,..., i}(n)=0$. Using a result of Garsia and Stanton, we deduce from our shelling for $\Delta (B_{2n})/S_2 \wr S_n$ that the ring of invariants $k[x_1,..., x_{2n}]^{S_2\wr S_n}$ is Cohen-Macaulay over any field $k$. | Lexicographic shellability for balanced complexes | 13,905 |
This paper verifies a conjecture of Edelman and Reiner regarding the homology of the $h$-complex of a Boolean algebra. A discrete Morse function with no low-dimensional critical cells is constructed, implying a lower bound on connectivity. This together with an Alexander duality result of Edelman and Reiner implies homology-vanishing also in high dimensions. Finally, possible generalizations to certain classes of supersolvable lattices are suggested. | Connectivity of h-complexes | 13,906 |
The Hadamard maximal determinant problem asks for the largest n-by-n determinant with entries in {+1,-1}. When n is congruent to 1 (mod 4), the maximal excess construction of Farmakis & Kounias has been the most successful general method for constructing large (though seldom maximal) determinants. For certain small n, however, still larger determinants have been known; several new records were recently reported in ArXiv preprint math.CO/0304410 . Here, we define ``3-normalized'' n-by-n Hadamard matrices, and construct large-determinant matrices of order n+1 from them. Our constructions account for most of the previous ``small n'' records, and set new records when n=37, 49, 65, 73, 77, 85, 93, and 97, most of which are beyond the reach of the maximal excess technique. We conjecture that our n=37 determinant, 72 x 9^{17} x 2^{36}, achieves the global maximum. | Large-determinant sign matrices of order 4k+1 | 13,907 |
The usual, or type A_n, Tamari lattice is a partial order on T_n^A, the triangulations of an (n+3)-gon. We define a partial order on T_n^B, the set of centrally symmetric triangulations of a (2n+2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the A_n Tamari lattice, and therefore that it deserves to be considered the B_n Tamari lattice. We define a bijection between T_n^B and the non-crossing partitions of type B_n defined by Reiner. For S any subset of [n], Reiner defined a pseudo-type BD^S_n, to which is associated a subset of the noncrossing partitions of type B_n. We show that the elements of T^B_n which correspond to the noncrossing partitions of type BD^S_n posess a lattice structure induced from their inclusion in T^B_n. | Tamari lattices and noncrossing partitions in type B and beyond | 13,908 |
In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n-2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that "cut across an ideal." Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P,I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular "generalized Bier spheres." In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P,I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields "many shellable spheres", most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres. | Bier spheres and posets | 13,909 |
We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan B'(K_n) of the graphical matroid of the complete graph K_n is homeomorphic to the space of phylogenetic trees T_n. | The Bergman complex of a matroid and phylogenetic trees | 13,910 |
We give a basis for the space V spanned by the lowest degree part \hat{s}_\lambda of the expansion of the Schur symmetric functions s_\lambda in terms of power sums, where we define the degree of the power sum p_i to be 1. In particular, the dimension of the subspace V_n spanned by those \hat{s}_\lambda for which \lambda is a partition of n is equal to the number of partitions of n whose parts differ by at least 2. We also show that a symmetric function closely related to \hat{s}_\lambda has the same coefficients when expanded in terms of power sums or augmented monomial symmetric functions. Proofs are based on the theory of minimal border strip decompositions of Young diagrams. | Bottom Schur functions | 13,911 |
We construct error correcting nonlinear binary codes using a construction of Bose and Chowla in additive number theory. Our method extends a construction of Graham and Sloane for constant weight codes. The new codes improve 1028 of the 7168 best known h-error correcting codes of wordlength at most 512 and h at most 14. We give assymptotical comparisons to shortened BCH codes. Tables of new lower bounds for "A(n,d)" are included. | Error correcting codes and B_h-sequences | 13,912 |
We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern overlaps introduced in this paper, which is a subgraph of the de Bruijn graph and which we prove to be Hamiltonian. In other cases we reduce a problem under consideration to known facts on unavoidable sets of words. We also give a relation between our problem and intensively studied universal cycles, and prove there exists a universal cycle for word patterns of any length over any alphabet. Keywords: pattern, word, (un)avoidability, de Bruijn graph, universal cycles | On unavoidable sets of word patterns | 13,913 |
Motivated by Stanley's results in \cite{St02}, we generalize the rank of a partition $\lambda$ to the rank of a shifted partition $S(\lambda)$. We show that the number of bars required in a minimal bar tableau of $S(\lambda)$ is max$(o, e + (\ell(\lambda) \mathrm{mod} 2))$, where $o$ and $e$ are the number of odd and even rows of $\lambda$. As a consequence we show that the irreducible projective characters of $S_n$ vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur's $Q_{\lambda}$ symmetric functions in terms of the power sum symmetric functions. | Minimal Bar Tableaux | 13,914 |
In this paper, we first give formulas for the order polynomial $\Omega (\Pw; t)$ and the Eulerian polynomial $e(\Pw; \lambda)$ of a finite labeled poset $(P, \omega)$ using the adjacency matrix of what we call the $\omega$-graph of $(P, \omega)$. We then derive various recursion formulas for $\Omega (\Pw; t)$ and $e(\Pw; \lambda)$ and discuss some applications of these formulas to Bernoulli numbers and Bernoulli polynomials. Finally, we give a recursive algorithm using a single linear operator on a vector space. This algorithm provides a uniform method to construct a family of new invariants for labeled posets $(\Pw)$, which includes the order polynomial $\Omega (\Pw; t)$ and the invariant $\tilde e(\Pw; \lambda) =\frac {e(\Pw; \lambda)}{(1-\lambda)^{|P|+1}}$. The well-known quasi-symmetric function invariant of labeled posets and a further generalization of our construction are also discussed. | A New Approach to Order Polynomials of Labeled Posets and Their
Generalizations | 13,915 |
Nested set complexes appear as the combinatorial core of De Concini-Procesi arrangement models. We show that nested set complexes are homotopy equivalent to the order complexes of the underlying meet-semilattices without their minimal elements. For atomic semilattices, we consider the realization of nested set complexes by simplicial fans as proposed in math.AG/0305142 by the first author and S. Yuzvinsky. We show that in this case the nested set complexes in fact are homeomorphic to the mentioned order complexes. | On the topology of nested set complexes | 13,916 |
Let $H$ be a fixed graph. A {\em fractional $H$-decomposition} of a graph $G$ is an assignment of nonnegative real weights to the copies of $H$ in $G$ such that for each $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$ in precisely one. An {\em $H$-packing} of a graph $G$ is a set of edge disjoint copies of $H$ in $G$. The following results are proved. For every fixed $k > 2$, every graph with $n$ vertices and minimum degree at least $n(1-1/9k^{10})+o(n)$ has a fractional $K_k$-decomposition and has a $K_k$-packing which covers all but $o(n^2)$ edges. | Asymptotically optimal $K_k$-packings of dense graphs via fractional
$K_k$-decompositions | 13,917 |
Let $X$ be $k$-regular graph on $v$ vertices and let $\tau$ denote the least eigenvalue of its adjacency matrix $A(X)$. If $\alpha(X)$ denotes the maximum size of an independent set in $X$, we have the following well known bound: \[ \alpha(X) \le\frac{v}{1-\frac{k}{\tau}}. \] It is less well known that if equality holds here and $S$ is a maximum independent set in $X$ with characteristic vector $x$, then the vector \[ x-\frac{|S|}{v}\one \] is an eigenvector for $A(X)$ with eigenvalue $\tau$. In this paper we show how this can be used to characterise the maximal independent sets in certain classes of graphs. As a corollary we show that a graph defined on the partitions of $\{1,...,9\}$ with three cells of size three is a core. | Independent sets in association schemes | 13,918 |
The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in ${\bf N}$, and standard Young tableaux by semistandard ones. For $r>0$, the Robinson-Schensted correspondence can be trivially extended, using the $r$-quotient map, to one between coloured permutations and pairs of standard $r$-ribbon tableaux built on a fixed $r$-core (the Stanton-White correspondence). This correspondence can also be generalised to arbitrary matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r$-core; the generalisation is derived from the RSK correspondence, again using the $r$-quotient map. Shimozono and White recently defined a more interesting generalisation of the Robinson-Schensted correspondence to coloured permutations and standard $r$-ribbon tableaux, one that (unlike the Stanton-White correspondence) respects the spin statistic (total height of ribbons) on standard $r$-ribbon tableaux, relating it directly to the colours of the coloured permutation. We define a construction establishing a bijective correspondence between general matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r$-core, which respects the spin statistic on those tableaux in a similar manner, relating it directly to the matrix entries. We also define a similar generalisation of the asymmetric RSK correspondence, in which case the matrix entries are taken from $\{0,1\}^r$. | Spin-preserving Knuth correspondences for ribbon tableaux | 13,919 |
We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let $G$ be a graph on $n$ vertices. A 2-lift of $G$ is a graph $H$ on $2n$ vertices, with a covering map $\pi:H \to G$. It is not hard to see that all eigenvalues of $G$ are also eigenvalues of $H$. In addition, $H$ has $n$ ``new'' eigenvalues. We conjecture that every $d$-regular graph has a 2-lift such that all new eigenvalues are in the range $[-2\sqrt{d-1},2\sqrt{d-1}]$ (If true, this is tight, e.g. by the Alon-Boppana bound). Here we show that every graph of maximal degree $d$ has a 2-lift such that all ``new'' eigenvalues are in the range $[-c \sqrt{d \log^3d}, c \sqrt{d \log^3d}]$ for some constant $c$. This leads to a polynomial time algorithm for constructing arbitrarily large $d$-regular graphs, with second eigenvalue $O(\sqrt{d \log^3 d})$. The proof uses the following lemma: Let $A$ be a real symmetric matrix such that the $l_1$ norm of each row in $A$ is at most $d$. Let $\alpha = \max_{x,y \in \{0,1\}^n, supp(x)\cap supp(y)=\emptyset} \frac {|xAy|} {||x||||y||}$. Then the spectral radius of $A$ is at most $c \alpha \log(d/\alpha)$, for some universal constant $c$. An interesting consequence of this lemma is a converse to the Expander Mixing Lemma. | Constructing expander graphs by 2-lifts and discrepancy vs. spectral gap | 13,920 |
It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n by n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem. The unimodality of this sequance, which follows, was conjectured by Stanley (1983). Several generalizations are given. | Ehrhart polynomials, simplicial polytopes, magic squares and a
conjecture of Stanley | 13,921 |
We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) $l$, we show that the numbers of regular and non-regular triangulations of $\Delta^l\times\Delta^k$ grow, respectively, as $k^{\Theta(k)}$ and $2^{\Omega(k^2)}$. For the special case of $l=2$, we relate triangulations to certain class of lozenge tilings. This allows us to compute the exact number of triangulations up to $k=15$, show that the number grows as $e^{\beta k^2/2 + o(k^2)}$ where $\beta\simeq 0.32309594$ and prove that the set of all triangulations is connected under geometric bistellar flips. The latter has as a corollary that the toric Hilbert scheme of the determinantal ideal of $2\times 2$ minors of a $3\times k$ matrix is connected, for every $k$. We include ``Cayley Trick pictures'' of all the triangulations of $\Delta^2\times \Delta^2$ and $\Delta^2\times \Delta^3$, as well as one non-regular triangulation of $\Delta^2\times \Delta^5$ and one of $\Delta^3\times \Delta^3$. | The Cayley trick and triangulations of products of simplices | 13,922 |
In this paper, we study the relations between the numerical structure of the optimal solutions of a convex programming problem defined on the edge set of a simple graph and the stability number (i.e. the maximum size of a subset of pairwise non-adjacent vertices) of the graph. Our analysis shows that the stability number of every graph G can be decomposed in the sum of the stability number of a subgraph containing a perfect 2-matching (i.e. a system of vertex-disjoint odd-cycles and edges covering the vertex-set) plus a term computable in polynomial time. As a consequence, it is possible to bound from above and below the stability number in terms of the matching number of a subgraph having a perfect 2-matching and other quantities computable in polynomial time. Our results are closely related to those by Lorentzen, Balinsky, Spielberg, and Pulleyblank on the linear relaxation of the vertex-cover problem. Moreover, The convex programming problem involved has important applications in information theory and extremal set theory where, as a graph capacity formula, has been used to answer a longstanding open question about qualitatively independet sets in the sense of Renyi (L. Gargano, J. K{\"o}rner, and U. Vaccaro, "Sperner capacities", Graphs and combinatorics, 9:31-46, 1993). | A Splitting Lemma | 13,923 |
In this note we point out the relation between Brion's formula for the lattice point generating function of a convex polytope in terms of the vertex cones [Brion1988] on the one hand, and the polar decomposition \`a la Lawrence/Varchenko [Lawrence1991, Varchenko1987] on the other. We then go on to prove a version of polar decomposition for non-simple polytopes. | Polar decomposition and Brion's theorem | 13,924 |
Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face vectors. A special class of multiplicial polytopes, also discovered by Bisztriczky, is comprised of the ordinary polytopes. These are a natural generalization of the cyclic polytopes. The flag vectors of ordinary polytopes are determined. This is used to give a surprisingly simple formula for the h-vector of the ordinary d-polytope with n+1 vertices and characteristic k: h_i=binom{k-d+i}{i}+(n-k)binom{k-d+i-1}{i-1}, for i at most d/2. In addition, a construction is given for 4-dimensional multiplicial polytopes having two-thirds of their vertices on a single facet, answering a question of Bisztriczky. | Flag vectors of multiplicial polytopes | 13,925 |
If the moments of a probability measure on $\R$ are interpreted as a specialization of complete homogeneous symmetric functions, its free cumulants are, up to sign, the corresponding specializations of a sequence of Schur positive symmetric functions $(f_n)$. We prove that $(f_n)$ is the Frobenius characteristic of the natural permutation representation of $\SG_n$ on the set of prime parking functions. This observation leads us to the construction of a Hopf algebra of parking functions, which we study in some detail. | A Hopf algebra of parking functions | 13,926 |
Let $G$ denote a near-polygon distance-regular graph with diameter $d\geq 3$, valency $k$ and intersection numbers $a_1>0$, $c_2>1$. Let $\theta_1$ denote the second largest eigenvalue for the adjacency matrix of $G$. We show $\theta_1$ is at most $(k-a_1-c_2)/(c_2-1)$. We show the following are equivalent: (i) Equality is attained above; (ii) $G$ is $Q$-polynomial with respect to $\theta_1$; (iii) $G$ is a dual polar graph or a Hamming graph. | An inequality for regular near polygons | 13,927 |
Let $G$ denote a distance-regular graph with diameter $D\ge 3$, valency $k$, and intersection numbers $a_i$, $b_i$, $c_i$. By a {\it pseudo cosine sequence} of $G$ we mean a sequence of real numbers $s_0, s_1, ..., s_D$ such that $s_0=1$ and $c_i s_{i-1}+a_i s_i+b_i s_{i+1}=k s_1 s_i$ for $ 0 \le i \le D-1$. Let $s_0, s_1,..., s_D$ and $p_0, p_1, ..., p_D$ denote pseudo cosine sequences of $G$. We say this pair of sequences is {\it tight} whenever $s_0 p_0, s_1 p_1, ..., s_D p_D$ is a pseudo cosine sequence of $G$. In this paper, we determine all the tight pairs of pseudo cosine sequences of $G$. | The Pseudo Cosine Sequences of a Distance-Regular Graph | 13,928 |
Alternating sign matrices with a U-turn boundary (UASMs) are a recent generalization of ordinary alternating sign matrices. Here we show that variations of these matrices are in bijective correspondence with certain symplectic shifted tableaux that were recently introduced in the context of a symplectic version of Tokuyama's deformation of Weyl's denominator formula. This bijection yields a formula for the weighted enumeration of UASMs. In this connection use is made of the link between UASMs and certain square ice configuration matrices. | U-turn Alternating Sign Matrices, Symplectic Shifted Tableaux and Their
Weighted Enumeration | 13,929 |
We show that for Bruhat intervals starting from the origin in simply-laced Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [e,u] only depend on the isomorphism type of [e,u]. | Invariance Combinatoire des Polynomes de Kazhdan-Lusztig sur les
intervalles partant de l'origine | 13,930 |
In this paper we present a unified approach to the spectral analysis of an hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well known formulas of Rodrigues type can be obtained directly. Moreover, other new Rodrigues type formulas come out when seeking for regular solutions of the associated functional equations. The main difference here is that, in contrast with the formulas appearing in the literature, we get non-ramified solutions which are useful for applications in combinatorics. Another fact, that becomes clear in this paper, is the role played by the theory of elliptic functions in the connection between ramified and non-ramified solutions. | A new approach to the theory of classical hypergeometric polynomials | 13,931 |
In this article we extend the idea of Turbo codes onto the Real Field. The channel is taken to result in block erasures and the only noise as being that due to quantization. The decoding in this case is reduced to reconstruction of the lost values. The encoding is done using critically sampled filter banks and introduction of an interleaver is found to reduce the mean square quantization error drastically. The permutation that gives the best recoverability is obtained in the 2 Channel case. Results are also obtained for M channel case. The algorithm for reconstruction of the lost values in the absence of quantization noise is obtained. | Turbo Codes over the Real Field | 13,932 |
Recently, Ehrenborg and Van Willenburg defined a class of bipartite graphs that correspond naturally to Ferrers diagrams, and proved several results about them. We give bijective proofs for the (already known) expressions for the number of spanning trees and (where applicable) Hamiltonian paths of these graphs. Their paper can be found at http://www.ms.uky.edu/~jrge/Papers/Ferrers_graphs.pdf . | Bijective Proofs for "Enumerative Properties of Ferrers Graphs" | 13,933 |
We study the determinant of the pxp circulant matrix whose first row is (1,-x,0,...,0,-y,0,...,0), the -y being in position q+1. The coefficients of this polynomial are integers that count certain classes of permutations. We show that all of the permutations that contribute to a fixed monomial x^ry^s have the same sign, and we determine that sign. We prove that a monomial x^ry^s appears if and only if p divides r+sq. Finally, we show that the size of the largest coefficient of the monomials that appear grows exponentially with p. We do this by proving that the permanent of the circulant whose first row is (1,1,0,...,0,1,0,...,0) is the sum of the absolute values of the coefficients of the monomials in the original determinant. | The combinatorics of a three-line circulant determinant | 13,934 |
Let G be a finite abelian group of order n. For a complex valued function f on G, let \fht denote the Fourier transform of f. The uncertainty inequality asserts that if f \neq 0 then |supp(f)| |supp(\fht)| \geq n. Answering a question of Terence Tao, the following improvement of the classical inequality is shown: Let d_1<d_2 be two consecutive divisors of n. If d_1 \leq k=|supp(f)| \leq d_2 then: |supp(\fht)| \geq \frac{n(d_1+d_2-k)}{d_1 d_2} | An uncertainty inequality for finite abelian groups | 13,935 |
A partition n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k is called non-squashing if p_1 + ... + p_j <= p_{j+1} for 1 <= j <= k-1. Hirschhorn and Sellers showed that the number of non-squashing partitions of n is equal to the number of binary partitions of n. Here we exhibit an explicit bijection between the two families, and determine the number of non-squashing partitions with distinct parts, with a specified number of parts, or with a specified maximal part. We use the results to solve a certain box-stacking problem. | On Non-Squashing Partitions | 13,936 |
In this paper, we generalize 2-trees by replacing triangles by quadrilaterals, pentagons or $k$-sided polygons ($k$-gons), where $k\geq 3$ is fixed. This generalization, to $k$-gonal 2-trees, is natural and is closely related, in the planar case, to some specializations of the cell-growth problem. Our goal is the labelled and unlabelled enumeration of $k$-gonal 2-trees according to the number $n$ of $k$-gons. We give explicit formulas in the labelled case, and, in the unlabelled case, recursive and asymptotic formulas. | Labelled and unlabelled enumeration of $k$-gonal 2-trees | 13,937 |
Two subspaces of a vector space are here called ``nonintersecting'' if they meet only in the zero vector. The following problem arises in the design of noncoherent multiple-antenna communications systems. How many pairwise nonintersecting M_t-dimensional subspaces of an m-dimensional vector space V over a field F can be found, if the generator matrices for the subspaces may contain only symbols from a given finite alphabet A subseteq F? The most important case is when F is the field of complex numbers C; then M_t is the number of antennas. If A = F = GF(q) it is shown that the number of nonintersecting subspaces is at most (q^m-1)/(q^{M_t}-1), and that this bound can be attained if and only if m is divisible by M_t. Furthermore these subspaces remain nonintersecting when ``lifted'' to the complex field. Thus the finite field case is essentially completely solved. In the case when F = C only the case M_t=2 is considered. It is shown that if A is a PSK-configuration, consisting of the 2^r complex roots of unity, the number of nonintersecting planes is at least 2^{r(m-2)} and at most 2^{r(m-1)-1} (the lower bound may in fact be the best that can be achieved). | Nonintersecting Subspaces Based on Finite Alphabets | 13,938 |
Let X(G) denote the flag complex of a graph G=(V,E) on n vertices. We study relations between the first eigenvalues of successive higher Laplacians of X(G). One consequence is the following result: Let \lambda_2(G) denote the second smallest eigenvalue of the Laplacian of G. If \lambda_2(G)> \frac{kn}{k+1} then the real k-th reduced cohomology group H^k(X(G)) is zero. Applications include a lower bound on the homological connectivity of the independent sets complex I(G), in terms of a new graph domination parameter \Gamma(G) defined via certain vector representations of G. This in turns implies a Hall type theorem for systems of disjoint representatives in hypergraphs. | Eigenvalues and homology of flag complexes and vector representations of
graphs | 13,939 |
We extend F.Pastjin's construction of uniform decomposable chains of finite rank to those of rank 'finite over a limit' and investigate infinite products and unions of such chains. We derive an extension of Pastjin's characterisation to uniform decomposable chains of small transfinite rank (less than or equal to 'omega+omega'). We conclude by indicating how the resulting descriptions can be iterated to chains of rank 'omega to the omega', and beyond. | Unifom chains of small transfinite rank | 13,940 |
We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the "splitting basis" for the homology of the partition lattice given in [Wa96], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in R^d. Let R_1,...,R_k be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles \rho_{R_i} in the homology of the proper part \bar{L_A} of the intersection lattice such that {\rho_{R_i}}_{i=1,...,k} is a basis for \tilde H_{d-2}(\bar{L_A}). This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements. | Geometrically constructed bases for homology of partition lattices of
types A, B and D | 13,941 |
In a recent paper [M], Mathon gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when $m\geq 5$ and $m\neq 9$, the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,1}-map is $(\floor {m/2} +1)$. This confirms our conjecture in [FLX]. For {p,q}-maps, we prove that if $m\geq 7$ and $m\neq 9$, then the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,q}-map is either $\floor {m/2} +1$ or $\floor{m/2} +2$. | On Mathon's construction of maximal arcs in Desarguesian planes. II | 13,942 |
We say a lattice tetrahedron whose centroid is its only non-vertex lattice point is lattice barycentric. The notation T(a,b,c) describes the lattice tetrahedron with vertices {0, e_1, e_2, a e_1 + b e_2 + c e_3}. Our result is that all such T(a,b,c) are unimodularly equivalent to T(3,3,4) or T(7,11,20). | On Lattice Barycentric Tetrahedra | 13,943 |
We consider planar lattice walks that start from (0,0), remain inthe first quadrant i, j >= 0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: -- they are counted by nice numbers (Kreweras 1965), -- the generating function of these numbers is algebraic (Gessel 1986), -- the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function (Flatto and Hahn 1984). These results are not well understood, and have been established via complicated proofs. Here we give a uniform derivation of all of them, whichis more elementary that those previously published.We then go further by computing the full law of the Markov chain. This helps to delimit the border of algebraicity: the associated probability generating function is no longer algebraic, unless a diagonal symmetry holds. Our proofs are based on the solution of certain functional equations,which are very simple to establish. Finding purely combinatorial proofs remains an open problem. | Walks in the quarter plane: Kreweras' algebraic model | 13,944 |
We introduce a monoid structure on the set of binary search trees, by a process very similar to the construction of the plactic monoid, the Robinson-Schensted insertion being replaced by the binary search tree insertion. This leads to a new construction of the algebra of Planar Binary Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric Functions and Free Symmetric Functions. We briefly explain how the main known properties of the Loday-Ronco algebra can be described and proved with this combinatorial point of view, and then discuss it from a representation theoretical point of view, which in turns leads to new combinatorial properties of binary trees. | The Algebra of Binary Search Trees | 13,945 |
We study the question of finding the maximal determinant of matrices of odd order with entries {-1,1}. The most general upper bound on the maximal determinant, due to Barba, can only be achieved when the order is the sum of two consecutive squares. It is conjectured that the bound is always attained in such cases. Apart from these, only in orders 3, 7, 9, 11, 17 and 21 has the maximal value been established. In this paper we confirm the results for these orders, and add order 15 to the list. We follow previous authors in exhaustively searching for candidate Gram matrices having determinant greater than or equal to the square of a known lower bound on the maximum. We then attempt to decompose each candidate as the product of a {-1,1}-matrix and its transpose. For order 15 we find four candidates, all of Ehlich block form, two having determinant (105*3^5*2^14)^2 and the others determinant (108*3^5*2^14)^2. One of the former decomposes (in an essentially unique way) while the remaining three do not. This result proves a conjecture made independently by W. D. Smith and J. H. E. Cohn. We also use our method to compute improved upper bounds on the maximal determinant in orders 29, 33, and 37, and to establish the range of the determinant function of {-1,1}-matrices in orders 9 and 11. | The maximal {-1,1}-determinant of order 15 | 13,946 |
A binomial coefficient identity due to Zhi-Wei Sun is the subject of half a dozen recent papers that prove it by various analytic techniques and establish a generalization. Here we give a simple proof that uses weight-reversing involutions on suitable configurations involving dominos and colorings. With somewhat more work, the method extends to the generalization also. | A combinatorial proof of Sun's "curious" identity | 13,947 |
We study generating functions for the number of involutions, even involutions, and odd involutions in $S_n$ subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the involution avoid another pattern $\tau$ or contain $\tau$ exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials of the second kind. | Involutions Restricted by 3412, Continued Fractions, and Chebyshev
Polynomials | 13,948 |
The tropical semiring (R, min, +) has enjoyed a recent renaissance, owing to its connections to mathematical biology as well as optimization and algebraic geometry. In this paper, we investigate the space of labeled n-point configurations lying on a tropical line in d-space, which is interpretable as the space of n-species phylogenetic trees. This is equivalent to the space of d by n matrices of tropical rank two, a simplicial complex. We prove that this simplicial complex is shellable for dimension d=3 and compute its homology in this case, conjecturing that this complex is shellable in general. We also investigate the space of d by n matrices of Barvinok rank two, a subcomplex directly related to optimization, giving a complete description of this subcomplex in the case d=3. | The moduli space of n tropically collinear points in R^d | 13,949 |
We compute the generating function of column-strict plane partitions with parts in {1,2,...,n}, at most c columns, p rows of odd length and k parts equal to n. This refines both, Krattenthaler's ["The major counting of nonintersecting lattice paths and generating functions for tableaux", Mem. Amer. Math. Soc. 115 (1995)] and the author's ["A method for proving polynomial enumeration formulas", preprint] refinement of the Bender-Knuth (ex-)Conjecture. The result is proved by an extension of the method for proving polynomial enumeration formulas which was introduced by the author to q-quasi-polynomials. | Another refinement of the Bender-Knuth (ex-)Conjecture | 13,950 |
Some enumerative aspects of the fans, called generalized associahedra, introduced by S. Fomin and A. Zelevinsky in their theory of cluster algebras are considered, in relation with a bicomplex and its two spectral sequences. A precise enumerative relation with the lattices of generalized noncrossing partitions is conjectured and some evidence is given. | Enumerative properties of generalized associahedra | 13,951 |
It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number $(G) of nested quantifiers in a such formula can serve as a measure for the ``first order complexity'' of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is \Theta(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log^* n, the inverse of the tower-function. The general picture, however, is still a mystery. | How Complex are Random Graphs in First Order Logic? | 13,952 |
It is well known that the numbers $(2m)! (2n)!/m! n! (m+n)!$ are integers, but in general there is no known combinatorial interpretation for them. When $m=0$ these numbers are the middle binomial coefficients $\binom{2n}{n}$, and when $m=1$ they are twice the Catalan numbers. In this paper, we give combinatorial interpretations for these numbers when $m=2$ or 3. | A Combinatorial Interpretation of The Numbers $6(2n)! /n! (n+2)!$ | 13,953 |
A convex triangular grid is represented by a planar digraph $G$ embedded in the plane so that (a) each bounded face is surrounded by three edges and forms an equilateral triangle, and (b) the union $\Rscr$ of bounded faces is a convex polygon. A real-valued function $h$ on the edges of $G$ is called a concave cocirculation if $h(e)=g(v)-g(u)$ for each edge $e=(u,v)$, where $g$ is a concave function on $\Rscr$ which is affinely linear within each bounded face of $G$. Knutson and Tao [J. Amer. Math. Soc. 12 (4) (1999) 1055--1090] proved an integrality theorem for so-called honeycombs, which is equivalent to the assertion that an integer-valued function on the boundary edges of $G$ is extendable to an integer concave cocirculation if it is extendable to a concave cocirculation at all. In this paper we show a sharper property: for any concave cocirculation $h$ in $G$, there exists an integer concave cocirculation $h'$ satisfying $h'(e)=h(e)$ for each boundary edge $e$ with $h(e)$ integer and for each edge $e$ contained in a bounded face where $h$ takes integer values on all edges. On the other hand, we explain that for a 3-side grid $G$ of size $n$, the polytope of concave cocirculations with fixed integer values on two sides of $G$ can have a vertex $h$ whose entries are integers on the third side but $h(e)$ has denominator $\Omega(n)$ for some interior edge $e$. Also some algorithmic aspects and related results on honeycombs are discussed. | Integer concave cocirculations and honeycombs | 13,954 |
We define a bijection that transforms an alternating sign matrix A with one -1 into a pair (N,E) where N is a (so called) ``neutral'' alternating sign matrix (with one -1) and E is an integer. The bijection preserves the classical parameters of Mills, Robbins and Rumsey as well as three new parameters (including E). It translates vertical reflection of A into vertical reflection of N. A hidden symmetry allows the interchange of E with one of the remaining two new parameters. A second bijection transforms (N,E) into a configuration of lattice paths called ``mixed configuration''. | Alternating sign matrices with one -1 under vertical reflection | 13,955 |
A Viterbi path of length n of a discrete Markov chain is a sequence of n+1 states that has the greatest probability of ocurring in the Markov chain. We divide the space of all Markov chains into Viterbi regions in which two Markov chains are in the same region if they have the same set of Viterbi paths. The Viterbi paths of regions of positive measure are called Viterbi sequences. Our main results are (1) each Viterbi sequence can be divided into a prefix, periodic interior, and suffix, and (2) as n increases to infinity (and the number of states remains fixed), the number of Viterbi regions remains bounded. The Viterbi regions correspond to the vertices of a Newton polytope of a polynomial whose terms are the probabilities of sequences of length n. We characterize Viterbi sequences and polytopes for two- and three-state Markov chains. | Viterbi Sequences and Polytopes | 13,956 |
For a graph $G$ whose degree sequence is $d_{1},..., d_{n}$, and for a positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph $H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $t_{1}(n,H)$ is twice the Tur\'{a}n number of $H$. In this paper we consider the case $p>1$. For some graphs $H$ we obtain exact results, for some others we can obtain asymptotically tight upper and lower bounds, and many interesting cases remain open. | A Turán Type Problem Concerning the Powers of the Degrees of a Graph
(revised) | 13,957 |
We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let \eta_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show that the fibers of \eta_K constitute the smallest lattice congruence with 1\equiv s for every s\in(S-K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order. | Lattice congruences of the weak order | 13,958 |
In this paper we show that the leading coefficient $\mu(y,w)$ of certain Kazhdan-Lusztig polynomials $P_{y,w}$ of the permutation group $\mathfrak S_n$ of 1,2,...,n are not greater than 1. More precisely, we show that the leading coefficients $\mu(y,w)$ are not greater than 1 whenever $a(y)< a(w)$, where $a: \mathfrak S_n\to\mathbf N$ is the function defined by Lusztig. | The leading coefficient of certain Kazhdan-Lusztig polynomials of the
permutation group $S_n$ | 13,959 |
We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern-avoidance. Applying these results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations. | Lattice congruences, fans and Hopf algebras | 13,960 |
For an arbitrary finite Coxeter group W we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B we obtain, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky's construction of the normal fan as a "cluster fan." Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two "Tamari" lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres. | Cambrian Lattices | 13,961 |
The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We also investigate the roots of Ehrhart polynomials. We prove that for fixed d, there exists a bounded region of C containing all roots of Ehrhart polynomials of d-polytopes, and that all real roots of these polynomials lie in [-d, [d/2]). In contrast, we prove that when the dimension d is not fixed the positive real roots can be arbitrarily large. We finish with an experimental investigation of the Ehrhart polynomials of cyclic polytopes and 0/1-polytopes. | Coefficients and Roots of Ehrhart Polynomials | 13,962 |
Pattern avoidance classes of permutations that cannot be expressed as unions of proper subclasses can be described as the set of subpermutations of a single bijection. In the case that this bijection is a permutation of the natural numbers a structure theorem is given. The structure theorem shows that the class is almost closed under direct sums or has a rational generating function. | Pattern avoidance classes and subpermutations | 13,963 |
A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was given by Gessel and Simion. Several extensions are given in this paper, including a recurrence formula, a specialization at roots of unity and type $B$ analogues. | Signed Mahonians | 13,964 |
The basic theme of this paper is the fact that if $A$ is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd\H os-Szemer\'edi [E-S]. (see also [El], [T], and [K-T] for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set $A + A$ small and then deriving that the product set $AA$ is large (using Freiman's structure theorem). (cf [N-T], [Na3].) We follow the reverse route and prove that if $|AA| < c|A|$, then $|A+A| > c^\prime |A|^2$ (see Theorem 1). A quantitative version of this phenomenon combined with Pl\"unnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in [E-S] on the growth in $k$. If $$ g(k) \equiv \text{min}\{|A[1]| + |A\{1\}|\} $$ over all sets $A\subset \Bbb Z$ of cardinality $|A| = k$ and where $A[1]$ (respectively, $A\{1\}$) refers to the simple sum (resp., product) of elements of $A$. (See (0.6), (0.7).) It was conjectured in [E-S] that $g(k)$ grows faster than any power of $k$ for $k\to\infty$. We will prove here that $\ell n g(k)\sim\frac{(\ell n k)^2}{\ell n \ell n k}$ (see Theorem 2) which is the main result of this paper. | The Erdős-Szemerédi problem on sum set and product set | 13,965 |
For a set of integers $I$, we define a $q$-ary $I$-cycle to be a assignment of the symbols 1 through $q$ to the integers modulo $q^n$ so that every word appears on some translate of $I$. This definition generalizes that of de Bruijn cycles, and opens up a multitude of questions. We address the existence of such cycles, discuss ``reduced'' cycles (ones in which the all-zeroes string need not appear), and provide general bounds on the shortest sequence which contains all words on some translate of $I$. We also prove a variant on recent results concerning decompositions of complete graphs into cycles and employ it to resolve the case of $|I|=2$ completely. | Generalized de Bruijn Cycles | 13,966 |
For an element $w$ in the Weyl algebra generated by $D$ and $U$ with relation $DU=UD+1$, the normally ordered form is $w=\sum c_{i,j}U^iD^j$. We demonstrate that the normal order coefficients $c_{i,j}$ of a word $w$ are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients $c_{i,j}$. We calculate the Weyl binomial coefficients: normal order coefficients of the element $(D+U)^n$ in the Weyl algebra. We extend all these results to the $q$-analogue of the Weyl algebra. We discuss further generalizations using $i$-rook numbers. | Rook numbers and the normal ordering problem | 13,967 |
We analyze the structure and enumerate Dumont permutations of the first and second kinds avoiding certain patterns or sets of patterns of length 3 and 4. Some cardinalities are given by Catalan numbers, powers of 2, little Schroeder numbers, and other known or related sequences. | Restricted Dumont permutations | 13,968 |
We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky. | Perfect Matchings and the Octahedron Recurrence | 13,969 |
In 1990, Kolesova, Lam and Thiel determined the 283,657 main classes of Latin squares of order 8. Using techniques to determine relevant Latin trades and integer programming, we examine representatives of each of these main classes and determine that none can contain a uniquely completable set of size less than 16. In three of these main classes, the use of trades which contain less than or equal to three rows, columns, or elements does not suffice to determine this fact. We closely examine properties of representatives of these three main classes. Writing the main result in Nelder's notation for critical sets, we prove that scs(8)=16. | The size of the smallest uniquely completable set in order 8 Latin
squares | 13,970 |
In 1998, Khodkar showed that the minimal critical set in the Latin square corresponding to the elementary abelian 2-group of order 16 is of size at most 124. Since the paper was published, improved methods for solving integer programming problems have been developed. Here we give an example of a critical set of size 121 in this Latin square, found through such methods. We also give a new upper bound on the size of critical sets of minimal size for the elementary abelian 2-group of order $2^n$: $4^{n}-3^{n}+4-2^{n}-2^{n-2}$. We speculate about possible lower bounds for this value, given some other results for the elementary abelian 2-groups of orders 32 and 64. An example of a critical set of size 29 in the Latin square corresponding to the elementary abelian 3-group of order 9 is given, and it is shown that any such critical set must be of size at least 24, improving the bound of 21 given by Donovan, Cooper, Nott and Seberry. | Critical sets in the elementary abelian 2- and 3- groups | 13,971 |
Working over the field of order 2 we consider those complete caps (maximal sets of points with no three collinear) which are disjoint from some codimension 2 subspace of projective space. We derive restrictive conditions which such a cap must satisfy in order to be complete. Using these conditions we obtain explicit descriptions of complete caps which do not meet every hyperplane in at least 5 points. In particular, we determine the set of cardinalities of all such complete caps in all dimensions. | Complete caps in projective space which are disjoint from a subspace of
codimension two | 13,972 |
Let $n,p,k$ be three positive integers. We prove that the rational fractions of $q$: $${n \brack k}_{q} {}_3\phi_{2} [ . {matrix}q^{1-k},q^{-p},q^{p-n} q,q^{1-n} {matrix}| q;q^{k+1}]\quad\textrm{and}\quad q^{(n-p)p}\qbi{n}{k}{q} {}_3\phi_2[ . {matrix}q^{1-k},q^{-p},q^{p-n} q,q^{1-n} {matrix}|q;q] $$ are polynomials of $q$ with positive integer coefficients. This generalizes a recent result of Lassalle (Ann. Comb. 6(2002), no. 3-4, 399-405), in the same way as the classical $q$-binomial coefficients refine the ordinary binomial coefficients. | Two new families of q-positive integers | 13,973 |
Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata's combinatorial interpretation of the Hermite polynomials as counting matchings of a set to obtain a triple lacunary generating function for the Hermite polynomials. We also give an umbral proof of this generating function. | A triple lacunary generating function for Hermite polynomials | 13,974 |
The blocker $A^{*}$ of an antichain $A$ in a finite poset $P$ is the set of elements minimal with the property of having with each member of $A$ a common predecessor. The following is done: 1. The posets $P$ for which $A^{**}=A$ for all antichains are characterized. 2. The blocker $A^*$ of a symmetric antichain in the partition lattice is characterized. 3. Connections with the question of finding minimal size blocking sets for certain set families are discussed. | A note on blockers in posets | 13,975 |
Applying a classical theorem of Smith, we show that the poset property of being Gorenstein$^*$ over $\mathbb{Z}_2$ is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. Incitti. We also show that the Bruhat order on the fixed points of an involutive automorphism induced by a Coxeter graph automorphism is isomorphic to the Bruhat order on the fixed subgroup viewed as a Coxeter group in its own right. | Fixed points of involutive automorphisms of the Bruhat order | 13,976 |
The recently introduced A-homotopy groups for graphs are investigated. The main concern of the present article is the construction of an infinite cell complex, the homotopy groups of which are isomorphic to the A-homotopy groups of the given graph. We present a natural candidate for such a cell complex, together with a homomorphism between the corresponding groups that indeed yields an isomorphism, if a cubical analog of the simplicial approximation theorem holds, which - so far - we were unable to prove. | A Homotopy Theory for Graphs | 13,977 |
An important problem in the theory of finite dynamical systems is to link the structure of a system with its dynamics. This paper contains such a link for a family of nonlinear systems over the field with two elements. For systems that can be described by monomials (including Boolean AND systems), one can obtain information about the limit cycle structure from the structure of the monomials. In particular, the paper contains a sufficient condition for a monomial system to have only fixed points as limit cycles. This condition depends on the cycle structure of the dependency graph of the system and can be verified in polynomial time. | Boolean Monomial Dynamical Systems | 13,978 |
The paper presents geometric models for the set WO of weak orders on a finite set. In particulary, WO is modeled as a set of vertices of a cubical subdivision of a permutahedron. This approach is an alternative to the usual representation of WO by means of weak order polytopes. | Weak order complexes | 13,979 |
The sandpile group of a connected graph is the group of recurrent configurations in the abelian sandpile model on this graph. We study the structure of this group for the case of regular trees. A description of this group is the following: Let T(d,h) be the d-regular tree of depth h and let V be the set of its vertices. Denote the adjacency matrix of T(d,h) by A and consider the modified Laplacian matrix D:=dI-A. Let the rows of D span the lattice L in Z^V. The sandpile group of T(d,h) is Z^V/L. We compute the rank, the exponent and the order of this abelian group and find a cyclic Hall-subgroup of order (d-1)^h. We find that the base (d-1)-logarithm of the exponent and of the order are asymptotically 3h^2/pi^2 and c_d(d-1)^h, respectively. We conjecture an explicit formula for the ranks of all Sylow subgroups. | On the sandpile group of regular trees | 13,980 |
Lin and Chang gave a generating function of convex polyominoes with an $m+1$ by $n+1$ minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is $$ \frac{m+n+mn}{m+n}{2m+2n\choose 2m}-\frac{2mn}{m+n}{m+n\choose m}^2. $$ We show that this result can be derived from some binomial coefficients identities related to the generating function of Jacobi polynomials. | The Number of Convex Polyominoes and the Generating Function of Jacobi
Polynomials | 13,981 |
The sequence of period 6 starting with 1, 1, 0, -1, -1, 0 appears in many different disguises in mathematics. Various q-versions of this sequence are found, and their relations with Euler's pentagonal numbers theorem and Chebyshev polynomials are discussed. | q-Analogs of classical 6-periodicity: from Euler to Chebyshev | 13,982 |
This paper studies structural aspects of lattice path matroids, a class of transversal matroids that is closed under taking minors and duals. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One of the main results is a characterization of lattice path matroids in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid. We examine some aspects related to key topics in the literature of transversal matroids and we determine the connectivity of lattice path matroids. We also introduce notch matroids, a minor-closed, dual-closed subclass of lattice path matroids, and we find their excluded minors. | Lattice Path Matroids: Structural Properties | 13,983 |
Following Penrose, we introduce a family of graph functions defined in terms of contractions of certain products of symmetric tensors along the edges of a graph. Special cases of these functions enumerate edge colorings and cycles of arbitrary length in graphs (in particular, Hamiltonian cycles). | The enumeration of edge colorings and Hamiltonian cycles by means of
symmetric tensors | 13,984 |
We consider the matrix ${\frak Z}_P=Z_P+Z_P^t$, where the entries of $Z_P$ are the values of the zeta function of the finite poset $P$. We give a combinatorial interpretation of the determinant of ${\frak Z}_P$ and establish a recursive formula for this determinant in the case in which $P$ is a boolean algebra. | Determinants Associated to Zeta Matrices of Posets | 13,985 |
Dreidel is a popular game played during the festival of Chanukah. Players start with an equal number of tokens, donate one token each to a common pot, and take turns spinning a four-sided top, called the dreidel. Depending on the side showing up, the spinner takes all the tokens in the pot, takes half the tokens in the pot, gives a token to the pot, or does nothing. Whenever the pot goes empty, everyone donates a token to the pot. The game continues till all players, except one, go broke. We prove that the expected length of a game of dreidel is at most quadratic in the number of tokens, irrespective of the number of players. This proves a conjecture of Doron Zeilberger. | Dreidel Lasts $O(N^2)$ Spins | 13,986 |
We construct a combinatorial model that is described by the cube recurrence, a nonlinear recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in $\mathbb{Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs. | The Cube Recurrence | 13,987 |
It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way. | One-Point Suspensions and Wreath Products of Polytopes and Spheres | 13,988 |
We show the first known example for a pattern $q$ for which $\lim_{n\to \infty} \sqrt[n]{S_n(q)}$ is not an integer. We find the exact value of the limit and show that it is irrational. Then we generalize our results to an infinite sequence of patterns. Finally, we provide further generalizations that start explaining why certain patterns are easier to avoid than others. Finally, we show that if $q$ is a layered pattern of length $k$, then $L(q)\geq (k-1)^2$ holds. | The limit of a Stanley-Wilf sequence is not always rational, and layered
patterns beat monotone patterns | 13,989 |
We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree $T$ on $n$ leaves with a path metric $d$, consider the pairwise distances $\{d(u,v)\}$ between leaves. It is well known that these determine the tree and the $d$ length of all edges. Here we consider distortions $\d$ of $d$ such that for all leaves $u$ and $v$ it holds that $|d(u,v) - \d(u,v)| < f/2$ if either $d(u,v) < M$ or $\d(u,v) < M$, where $d$ satisfies $f \leq d(e) \leq g$ for all edges $e$. Given such distortions we show how to reconstruct in polynomial time a forest $T_1,...,T_{\alpha}$ such that the true tree $T$ may be obtained from that forest by adding $\alpha-1$ edges and $\alpha-1 \leq 2^{-\Omega(M/g)} n$. Metric distortions arise naturally in phylogeny, where $d(u,v)$ is defined by the log-det of a covariance matrix associated with $u$ and $v$. of a covariance matrix associated with $u$ and $v$. When $u$ and $v$ are ``far'', the entries of the covariance matrix are small and therefore $\d(u,v)$, which is defined by log-det of an associated empirical-correlation matrix may be a bad estimate of $d(u,v)$ even if the correlation matrix is ``close'' to the covariance matrix. Our metric results are used in order to show how to reconstruct phylogenetic forests with small number of trees from sequences of length logarithmic in the size of the tree. Our method also yields an independent proof that phylogenetic trees can be reconstructed in polynomial time from sequences of polynomial length under the standard assumptions in phylogeny. Both the metric result and its applications to phylogeny are almost tight. | Distorted metrics on trees and phylogenetic forests | 13,990 |
We give a Newton type rational interpolation formula (Theorem \ref{theo}). It contains as a special case the original Newton interpolation, as well as the recent interpolation formula of Zhi-Guo Liu, which allows to recover many important classical $q$-series identities. We show in particular that some bibasic identities are a consequence of our formula. | Rational Interpolation and Basic Hypergeometric Series | 13,991 |
In this paper, a result of Albert, Atkinson, Handley, Holton, and Stromquist [Electron. J. Combin. 9 (2002), #R5] which characterizes the optimal packing behavior of the pattern 1243 is generalized in two directions. The packing densities of layered patterns of type (1^a,a) and (1,1,b) are computed. | Optimal Packing Behavior of some 2-block Patterns | 13,992 |
We show that the left-greedy algorithm is a better algorithm than the right-greedy algorithm for sorting permutations using t stacks in series when t>1. We also supply a method for constructing some permutations that can be sorted by t stacks in series and from this get a lower bound on the number of permutations of length n that are sortable by t stacks in series. Finally we show that the left-greedy algorithm is neither optimal nor defines a closed class of permutations for t>2. | Comparing algorithms for sorting with t stacks in series | 13,993 |
The property of balance (in the sense of Feder and Mihail) is investigated in the context of paving matroids. The following examples are exhibited: (a) a class of ``sparse'' paving matroids that are balanced, but at the same time rich enough combinatorially to permit the encoding of hard counting problems; and (b) a paving matroid that is not balanced. The computational significance of (a) is the following. As a consequence of balance, there is an efficient algorithm for approximating the number of bases of a sparse paving matroid within specified relative error. On the other hand, determining the number of bases exactly is likely to be computationally intractable. | Two remarks concerning balanced matroids | 13,994 |
Applying the enumeration of sparse set partitions, we show that the number of set systems H such that the emptyset is not in H, the total cardinality of edges in H is n, and the vertex set of H is {1, 2, ..., m}, equals (1/log(2)+o(1))^nb_n where b_n is the n-th Bell number. The same asymptotics holds if H may be a multiset. If vertex degrees in H are restricted to be at most k, the asymptotics is (1/alpha_k+o(1))^nb_n where alpha_k is the unique root of x^k/k!+...+x^1/1!-1 in (0,1]. | Counting set systems by weight | 13,995 |
We undertake a combinatorial study of the piecewise linear map g : R^{2m+2n} --> R^{mn} which assigns to the four vectors a, A in R^m and b, B in R^n the m by n matrix given by g_{ij} = min (a_i + b_j, A_i+B_j). This map arises naturally in Pachter and Sturmfels's work on the tropical geometry of statistical models. The image of g has been a subject of recent interest; it is the positive part of the tropical algebraic variety which parameterizes n-tuples of points on a tropical line in m-space. The domains of linearity of g are the regions of the real hyperplane arrangement A_{m,n}, corresponding to the complete bipartite graph K_{m,n}. We explain how the images of (some of) the regions provide two polyhedral subdivisions of the image of g, one of which is a refinement of the other. The finer subdivision is particularly nice enumeratively: it has 2 {m \choose 2} {n \choose 2} r_{m-2,n-2} maximum-dimensional cells, where r_{m-2,n-2} is the number of regions of the arrangement A_{m-2,n-2}. | A tropical morphism related to the hyperplane arrangement of the
complete bipartite graph | 13,996 |
The generating functions of the major index and of the flag-major index, with each of the one-dimensional characters over the symmetric and hyperoctahedral group, respectively, have simple product formulas. In this paper, we give a factorial-type formula for the generating function of the D-major index with sign over the Weyl groups of type D. This completes a picture which is now known for all the classical Weyl groups. | Signed Mahonian polynomials for classical Weyl groups | 13,997 |
We report about some results, interesting examples, problems and conjectures revolving around the parabolic Kostant partition functions, the parabolic Kostka polynomials and ``saturation'' properties of several generalizations of the Littlewood--Richardson numbers. | An Invitation to the Generalized Saturation Conjecture | 13,998 |
Various statistics on wreath products are defined via canonical words, "colored" right to left minima and "colored" descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of binomial-Stirling type. These extended Stirling numbers determine (via matrix inversion) dual systems, which are also shown to have combinatorial realizations within the wreath product. The above setting also gives rise to MacMahon type equi-distribution theorem over subsets with prescribed statistics. | Statistics on Wreath Products and Generalized Binomial-Stirling Numbers | 13,999 |
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