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We characterize the Sheffer sequences by a single convolution identity $$ F^{(y)} p_{n}(x) = \sum _{k=0}^{n}\ p_{k}(x)\ p_{n-k}(y)$$ where $F^{(y)}$ is a shift-invariant operator. We then study a generalization of the notion of Sheffer sequences by removing the requirement that $F^{(y)}$ be shift-invariant. All these solutions can then be interpreted as cocommutative coalgebras. We also show the connection with generalized translation operators as introduced by Delsarte. Finally, we apply the same convolution to symmetric functions where we find that the ``Sheffer'' sequences differ from ordinary full divided power sequences by only a constant factor. | A simpler characterization of Sheffer polynomial | 13,300 |
We define the Artinian and Noetherian algebra which consist of formal series involving exponents which are not necessarily integers. All of the usual operations are defined here and characterized. As an application, we compute the algebra of symmetric functions with nonnegative real exponents. The applications to logarithmic series and the Umbral calculus are deferred to another paper. On d\'efinit ici les alg\`ebres Artinienne et Noetherienne comme \'etant des alg\`ebres constitu\'ees des s\'eries formelles \`a exposants pas n\'ecessairement entiers. On definit sur ces alg\`ebres toutes les op\'erations classiques et on les caracterise. Comme exemple d'exploitation de cette th\'eorie, on s'interesse \`a alg\`ebre de fonctions sym\'etriques à exponsants r\`eels en nonn\'egatifs. Une autre publication est consacr\'ee aux applications aux series logarithmiques et au calcul ombral. | Series with general exponents | 13,301 |
We generalize the Stirling numbers of the first kind $s(a,k)$ to the case where $a$ may be an arbitrary real number. In particular, we study the case in which $a$ is an integer. There, we discover new combinatorial properties held by the classical Stirling numbers, and analogous properties held by the Stirling numbers $s(n,k)$ with $n$ a negative integer. On g\'{e}n\'{e}ralise ici les nombres de Stirling du premier ordre $s(a,k)$ au cas o\`u $a$ est un r\'eel quelconque. On s'interesse en particulier au cas o\`u $a$ est entier. Ceci permet de mettre en evidence de nouvelles propri\'et\'es combinatoires aux quelles obeissent les nombres de Stirling usuels et des propri\'et\'es analougues auquelles obeissent les nombres de Stirling $s(n,k)$ o\`u $n$ est un entier n\`egatif. | A generalization of Stirling numbers | 13,302 |
We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. On cherche ici \`a d\'eterminer est la meilleure g\'en\'eralisation possible des factorielles et des coefficients du bin\^oome. On s'interesse \`a plusieurs exemples, \`a leurs propri\'et\'es combinatoires, et aux differentes relations qu'ils mettent en jeu. | A generalization of the binomial coefficients | 13,303 |
We generalize the Umbral Calculus of G-C. Rota by studying not only sequences of polynomials and inverse power series, or even the logarithms studied in, but instead we study sequences of formal expressions involving the iterated logarithms and x to an arbitrary real power. Using a theory of formal power series with real exponents, and a more general definition of factorial, binomial coefficient, and Stirling numbers to all the real numbers, we define the Iterated Logarithmic Algebra I. Its elements are the formal representations of the asymptotic expansions of a large class of real functions, and we define the harmonic logarithm basis of I which will be interpreted as a generalization of the powers x^n since it behaves nicely with respect to the derivative We classify all operators over I which commute with the derivative (classically these are known as shift-invariant operators), and formulate several equivalent definitions of a sequence of binomial type. We then derive many formulas useful towards the calculation of these sequences including the Recurrence Formula, the Transfer Formula, and the Lagrange Inversion Formula. Finally, we study Sheffer sequences, and give many examples. | The iterated logarithmic algebra | 13,304 |
An extension of the theory of the Iterated Logarithmic Algebra gives the logarithmic analog of a Sheffer or Appell sequence of polynomials. This leads to several examples including Stirling's formula and a logarithmic version of the Euler-MacLaurin summation formula. Gr\^ace \`a une g\'en\'eralisation de la th\'eorie de l'alg\`ebre des logarithmes it\'er\'es, on definit un analogue logarithmique des suites de polyn\^omes de Sheffer et d'Appell. Quelques exemples d'applications permettent de d\'eduire la formule de Stirling ainsi qu'un version logarithmique de la formule de sommation de Euler--MacLaurin. | The iterated logarithmic algebra II: Sheffer sequences | 13,305 |
We take advantage of the combinatorial interpretations of many sequences of polynomials of binomial type to define a sequence of symmetric functions corresponding to each sequence of polynomials of binomial type. We derive many of the results of Umbral Calculus in this context including a Taylor's expansion and a binomial identity for symmetric functions. Surprisingly, the delta operators for all the sequences of binomial type correspond to the same operator on symmetric functions. On s'appuie ici sur les interpr\'etations combinatoires de nombreuses suites de polyn\^omes de type binomial pour d\'efinir une suite de fonctions sym\'etriques associ\'ee \`a chque suite de polyn\^omes de type binomial. On retrouve dans ce cadre, de nombreaux r\'esultats du calcul ombral, en particulier une version de la formule de Taylor et la formule d'identit\'e du bin\^ome pour les fonctions sym\'etriques. On s'aper\oit que les op\'erateurs differentiels de degr\'e un pour toutes les suite de polyn\^omes de type a binomial correspondent \`a un op\'erateur unique sur les fonction sym\'etriques. | Sequences of symmetric functions of binomial type | 13,306 |
A Richman game is a combinatorial game in which, rather than alternating moves, the two players bid for the privilege of making the next move. We consider both the case where the players pay each other and the case where the players pay a neutral third party. We find optimal strategies considering both the case where the players know how much money their opponent has and the case where they do not. | Richman games | 13,307 |
New proofs are given for Monjardet's theorem that all strong simple games (i.e., ipsodual elements of the free distributive lattice) can be generated by the median operation. Tighter limits are placed on the number of iterations necessary. Comparison is drawn with the $\chi$ function which also generates all strong simple games. | A new proof of Monjardet's median theorem | 13,308 |
We review the Green/Kleitman/Leeb interpretation of de Bruijn's symmetric chain decomposition of ${\cal B}_{n}$, and explain how it can be used to find a maximal collection of disjoint symmetric chains in the nonsymmetric lattice of partitions of a set. | Symmetric chain decompositions of B_n and Pi_n | 13,309 |
Certain endgame considerations in the two-player Nigerian Mancala-type game Ayo can be identified with the problem of finding winning positions in the solitaire game Tchoukaitlon. The periodicity of the pit occupancies in $s$ stone winning positions is determined. Given $n$ pits, the number of stones in a winning position is found to be asymptotically bounded by $n^{2}/\pi$. | The combinatorics of Mancala-type games: Ayo, Tchoukaitlon, and 1/pi | 13,310 |
Finite obstruction sets for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. It has been known for several years that, in principle, obstruction sets can be mechanically computed for most natural lower ideals. In this paper, we describe a general-purpose method for finding obstructions by using a bounded treewidth (or pathwidth) search. We illustrate this approach by characterizing certain families of cycle-cover graphs based on the two well-known problems: $k$-{\sc Feedback Vertex Set} and $k$-{\sc Feedback Edge Set}. Our search is based on a number of algorithmic strategies by which large constants can be mitigated, including a randomized strategy for obtaining proofs of minimality. | Obstructions to within a few vertices or edges of acyclic | 13,311 |
A set partition technique that is useful for identifying wires in cables can be recast in the language of 0--1 matrices, thereby resolving an open problem stated by R.~L. Graham in Volume 1 of this journal. The proof involves a construction of 0--1 matrices having row and column sums without gaps. | The Knowlton-Graham partition problem | 13,312 |
A $(v,k,t)$ {\em covering design}, or {\em covering}, is a family of $k$-subsets, called blocks, chosen from a $v$-set, such that each $t$-subset is contained in at least one of the blocks. The number of blocks is the covering's {\em size}, and the minimum size of such a covering is denoted by $C(v,k,t)$. This paper gives three new methods for constructing good coverings: a greedy algorithm similar to Conway and Sloane's algorithm for lexicographic codes~\cite{lex}, and two methods that synthesize new coverings from preexisting ones. Using these new methods, together with results in the literature, we build tables of upper bounds on $C(v,k,t)$ for $v \leq 32$, $k \leq 16$, and $t \leq 8$.% | New constructions for covering designs | 13,313 |
Aim: Present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Ascend from Nim to chess in small strides at a gradient that's not too steep. Presentation: Informal; examples of games sampled from various strategic viewing points along scenic mountain trails, which illustrate the theory. | Scenic trails ascending from sea-level Nim to alpine chess | 13,314 |
The ``losing positions" of certain combinatorial games constitute linear error detecting and correcting codes. We show that a large class of games that can be cast in the form of *annihilation games*, provides a potentially polynomial method for computing codes (*anncodes*). We also give a short proof of the basic properties of the previously known *lexicodes*, which are defined by means of an exponential algorithm, and are related to game theory. The set of lexicodes is seen to constitute a subset of the set of anncodes. In the final section we indicate, by means of an example, how the method of producing lexicodes can be applied optimally to find anncodes. Some extensions are indicated. | Error-correcting codes derived from combinatorial games | 13,315 |
Cayley graph techniques are introduced for the problem of constructing networks having the maximum possible number of nodes, among networks that satisfy prescribed bounds on the parameters maximum node degree and broadcast diameter. The broadcast diameter of a network is the maximum time required for a message originating at a node of the network to be relayed to all other nodes, under the restriction that in a single time step any node can communicate with only one neighboring node. For many parameter values these algebraic methods yield the largest known constructions, improving on previous graph-theoretic approaches. It has previously been shown that hypercubes are optimal for degree $k$ and broadcast diameter $k$. A construction employing dihedral groups is shown to be optimal for degree $k$ and broadcast diameter $k+1$. | Algebraic constructions of efficient broadcast networks | 13,316 |
The results of computer searches for large graphs with given (small) degree and diameter are presented. The new graphs are Cayley graphs of semidirect products of cyclic groups and related groups. One fundamental use of our ``dense graphs'' is in the design of efficient communication network topologies. | New results for the degree/diameter problem | 13,317 |
In previous paper, the author applied the permanent-determinant method of Kasteleyn and its non-bipartite generalization, the Hafnian-Pfaffian method, to obtain a determinant or a Pfaffian that enumerates each of the ten symmetry classes of plane partitions. After a cosmetic generalization of the Kasteleyn method, we identify the matrices in the four determinantal cases (plain plane partitions, cyclically symmetric plane partitions, transpose-complement plane partitions, and the intersection of the last two types) in the representation theory of sl(2,C). The result is a unified proof of the four enumerations. | Four symmetry classes of plane partitions under one roof | 13,318 |
A digraph $D$ is called {\bf noneven} if it is possible to assign weights of 0,1 to its arcs so that $D$ contains no cycle of even weight. A noneven digraph $D$ corresponds to one or more nonsingular sign patterns. Given an $n \times n$ sign pattern $H$, a {\bf symplectic pair} in $Q(H)$ is a pair of matrices $(A,D)$ such that $A \in Q(H)$, $D \in Q(H)$, and $A^T D = I$. An unweighted digraph $D$ allows a matrix property $P$ if at least one of the sign patterns whose digraph is $D$ allows $P$. Thomassen characterized the noneven, 2-connected symmetric digraphs (i.e., digraphs for which the existence of arc $(u,v)$ implies the existence of arc $(v ,u))$. In the first part of our paper, we recall this characterization and use it to determine which strong symmetric digraphs allow symplectic pairs. A digraph $D$ is called {\bf semi-complete} if, for each pair of distinct vertices $(u,v)$, at least one of the arcs digraph. In the second part of our paper, we fill a gap in these two characterizations and present and prove correct versions of the main theorems involved. We then pr oceed to determine which digraphs from these classes allow symplectic pairs. $(u,v)$ and $(v,u)$ exists in $D$. Thomassen presented a characterization of two classes of strong, noneven digraphs: the semi-complete class and the class of digraphs for which each vertex has total degree which exceeds or equals the size of the digraph. In the second part of our paper, we fill a gap in these two characterizations and present and prove correct versions of the main theorems involved. We then p oceed to determine which digraphs from these classes allow symplectic pairs. | On non-even digraphs and symplectic pairs | 13,319 |
The canonization theorem says that for given m,n for some m^* (the first one is called ER(n;m)) we have: for every function f with domain [{1, ...,m^*}]^n, for some A in [{1, ...,m^*}]^m, the question of when the equality f({i_1, ...,i_n})=f({j_1, ...,j_n}) (where i_1< ... <i_n and j_1 < ... < j_n are from A) holds has the simplest answer: for some v subseteq {1, ...,n} the equality holds iff (for all l in v)(i_l = j_l). In this paper we improve the bound on ER(n,m) so that fixing n the number of exponentiation needed to calculate ER(n,m) is best possible. | Finite canonization | 13,320 |
A (v,k,t) covering design, or covering, is a family of k-subsets, called blocks, chosen from a v-set, such that each t-subset is contained in at least one of the blocks. The number of blocks is the covering's size}, and the minimum size of such a covering is denoted by C(v,k,t). It is easy to see that a covering must contain at least (v choose t)/(k choose t) blocks, and in 1985 R\"odl [European J. Combin. 5 (1985), 69-78] proved a long-standing conjecture of Erd\H{o}s and Hanani [Publ. Math. Debrecen 10 (1963), 10-13] that for fixed k and t, coverings of size (v choose t)/(k choose t) (1+o(1)) exist (as v \to \infty). An earlier paper by the first three authors [J. Combin. Des. 3 (1995), 269-284] gave new methods for constructing good coverings, and gave tables of upper bounds on C(v,k,t) for small v, k, and t. The present paper shows that two of those constructions are asymptotically optimal: For fixed k and t, the size of the coverings constructed matches R\"odl's bound. The paper also makes the o(1) error bound explicit, and gives some evidence for a much stronger bound. | Asymptotically optimal covering designs | 13,321 |
Let $P_1, P_2,\ldots, P_{d+1}$ be pairwise disjoint $n$-element point sets in general position in $d$-space. It is shown that there exist a point $O$ and suitable subsets $Q_i\subseteq P_i \; (i=1, 2, \ldots, d+1)$ such that $|Q_i|\geq c_d|P_i|$, and every $d$-dimensional simplex with exactly one vertex in each $Q_i$ contains $O$ in its interior. Here $c_d$ is a positive constant depending only on $d$. | A Tverberg-type result on multicolored simplices | 13,322 |
Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order $n$ equals $A(n):={{1!4!7! ... (3n-2)!} \over {n!(n+1)! ... (2n-1)!}}$. Mills, Robbins, and Rumsey also made the stronger conjecture that the number of such matrices whose (unique) `1' of the first row is at the $r^{th}$ column, equals $A(n) {{n+r-2} \choose {n-1}}{{2n-1-r} \choose {n-1}}/ {{3n-2} \choose {n-1}}$. Standing on the shoulders of A.G. Izergin, V. E. Korepin, and G. Kuperberg, and using in addition orthogonal polynomials and $q$-calculus, this stronger conjecture is proved. | Proof of the Refined Alternating Sign Matrix Conjecture | 13,323 |
This expository note presents simplifications of a theorem due to Gy\H{o}ri and an algorithm due to Franzblau and Kleitman: Given a family $F$ of $m$ intervals on a linearly ordered set of $n$ elements, we can construct in $O(m+n)^2$ steps an irredundant subfamily having maximum cardinality, as well as a generating family having minimum cardinality. The algorithm is of special interest because it solves a problem analogous to finding a maximum independent set, but on a class of objects that is more general than a matroid. This note is also a complete, runnable computer program, which can be used for experiments in conjunction with the public-domain software of {\sl The Stanford GraphBase}. | Irredundant intervals | 13,324 |
We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is of multiplicity $k$. More generally, we describe the homotopy type of the one-point compactification of strata in the boundary of the set of hyperbolic polynomials, that are defined via certain restrictions on root multiplicities, by order complexes of posets of compositions. In general, the methods are combinatorial and the topological problems are mostly reduced to the study of partially ordered sets. | Combinatorics and topology of stratifications of the space of monic
polynomials with real coefficients | 13,325 |
We show that the class of trapezoid orders in which no trapezoid strictly contains any other trapezoid strictly contains the class of trapezoid orders in which every trapezoid can be drawn with unit area. This is different from the case of interval orders, where the class of proper interval orders is exactly the same as the class of unit interval orders. | Proper and Unit Trapezoid Orders and Graphs | 13,326 |
A Hamiltonian graph $G$ of order $n$ is $k$-ordered, $2\leq k \leq n$, if for every sequence $v_1, v_2, \ldots ,v_k$ of $k$ distinct vertices of $G$, there exists a Hamiltonian cycle that encounters $v_1, v_2, \ldots , v_k$ in this order. In this paper, answering a question of Ng and Schultz, we give a sharp bound for the minimum degree guaranteeing that a graph is a $k$-ordered Hamiltonian graph under some mild restrictions. More precisely, we show that there are $\varepsilon, n_0> 0$ such that if $G$ is a graph of order $n\geq n_0$ with minimum degree at least $\lceil \frac{n}{2} \rceil + \lfloor \frac{k}{2} \rfloor - 1$ and $2\leq k \leq \eps n$, then $G$ is a $k$-ordered Hamiltonian graph. It is also shown that this bound is sharp for every $2\leq k \leq \lfloor \frac{n}{2} \rfloor$. | On $k$-ordered Hamiltonian Graphs | 13,327 |
Recently we have developed a new method in graph theory based on the Regularity Lemma. The method is applied to find certain spanning subgraphs in dense graphs. The other main general tool of the method, beside the Regularity Lemma, is the so-called Blow-up Lemma. This lemma helps to find bounded degree spanning subgraphs in $\varepsilon$-regular graphs. Our original proof of the lemma is not algorithmic, it applies probabilistic methods. In this paper we provide an algorithmic version of the Blow-up Lemma. The desired subgraph, for an $n$-vertex graph, can be found in time $O(nM(n))$, where $M(n)=O(n^{2.376})$ is the time needed to multiply two $n$ by $n$ matrices with 0,1 entries over the integers. We show that the algorithm can be parallelized and implemented in $NC^5$. | An Algorithmic Version of the Blow-up Lemma | 13,328 |
Consider the question: Given integers $k<d<n$, does there exist a simple $d$-polytope with $n$ faces of dimension $k$? We show that there exist numbers $G(d,k)$ and $N(d,k)$ such that for $n> N(d,k)$ the answer is yes if and only if $n\equiv 0\quad \pmod {G(d,k)}$. Furthermore, a formula for $G(d,k)$ is given, showing that e.g. $G(d,k)=1$ if $k\ge \left\lfloor\frac{d+1}{2}\right\rfloor$ or if both $d$ and $k$ are even, and also in some other cases (meaning that all numbers beyond $N(d,k)$ occur as the number of $k$-faces of some simple $d$-polytope). This question has previously been studied only for the case of vertices ($k=0$), where Lee \cite{Le} proved the existence of $N(d,0)$ (with $G(d,0)=1$ or $2$ depending on whether $d$ is even or odd), and Prabhu \cite{P2} showed that $N(d,0) \le cd\sqrt {d}$. We show here that asymptotically the true value of Prabhu's constant is $c=\sqrt2$ if $d$ is even, and $c=1$ if $d$ is odd. | The number of faces of a simple polytope | 13,329 |
We show the nonequivalence of combinations of several natural geometric restrictions on trapezoid representations of trapezoid orders. Each of the properties unit parallelogram, unit trapezoid and proper parallelogram, unit trapezoid and parallelogram, unit trapezoid, proper parallelogram, proper trapezoid and parallelogram, proper trapezoid, parallelogram, and trapezoid is shown to be distinct from each of the others. Additionally, interval orders are shown to be both unit trapezoid and proper parallelogram orders. | Trapezoid Order Classification | 13,330 |
Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement can be realized as monotone path polytopes. The construction is an extension of that of the permutahedron and yields interesting connections between enumerative combinatorics of hyperplane arrangements and geometry of monotone path polytopes. | Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements | 13,331 |
We consider a specialization $Y_M(q,t)$ of the Tutte polynomial of a matroid $M$ which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of $M$. We show that the coefficients of $Y_M(1-p,t)$ are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independent set complexes of the duals of the truncations of $M$. In particular, we obtain a new homological interpretation for the coefficients of the characteristic polynomial of a matroid. | The Tutte dichromate and Whitney homology of matroids | 13,332 |
Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating function $H(x)$ of all 1342-avoiding permutations of length $n$ as well as an {\em exact} formula for their number $S_n(1342)$. While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of length $n$ equals that of labeled plane trees of a certain type on $n$ vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, $H(x)$ turns out to be algebraic, proving the first nonmonotonic, longer-than-three instance of a conjecture of Zeilberger and Noonan. We also prove that $\sqrt[n]{S_n(1342)}$ converges to 8, so in particular, $lim_{n\rightarrow \infty}(S_n(1342)/S_n(1234))=0$. | Exact enumeration of 1342-avoiding permutations: A close link with
labeled trees and planar maps | 13,333 |
The Shi arrangement ${\mathcal S}_n$ is the arrangement of affine hyperplanes in ${\mathbb R}^n$ of the form $x_i - x_j = 0$ or $1$, for $1 \leq i < j \leq n$. It dissects ${\mathbb R}^n$ into $(n+1)^{n-1}$ regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of ${\mathcal S}_n$ containing the hyperplanes $x_i - x_j = 0$ and to the extended Shi arrangements. | A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes | 13,334 |
It has been shown that there is a Hamilton cycle in every connected Cayley graph on each group G whose commutator subgroup is cyclic of prime-power order. This paper considers connected, vertex-transitive graphs X of order at least 3 where the automorphism group of X contains a transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs, only the Petersen graph is not hamiltonian. | Automorphism groups with cyclic commutator subgroup and Hamilton cycles | 13,335 |
We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself. | On non-Hamiltonian circulant digraphs of outdegree three | 13,336 |
We give a combinatorial proof of the result of Hetyei and Reiner that there are exactly $n!/3$ permutations of length $n$ in the minmax tree representation of which the $i$th node is a leaf. We also prove the new result that the number of $n$-permutations in which this node has one child is $n!/3$ as well, implying that the same holds for those in which this node has two children. | A Combinatorial proof of a result of Hetyei and Reiner on Foata-Strehl
type permutation trees | 13,337 |
Let $P$ and $Q$ be bounded posets. In this note, a lemma is introduced that provides a set of sufficient conditions for the proper part of $P$ being homotopy equivalent to the suspension of the proper part of~$Q$. An application of this lemma is a unified proof of the sphericity of the higher Bruhat orders under both inclusion order (a known proved earlier by Ziegler) and single step inclusion order (which was not previously known). | A Suspension Lemma for Bounded Posets | 13,338 |
There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic $d$ polytope with $n$ vertices. In this paper it is shown that both of them have the homotopy type of a sphere of dimension $n-d-3$. Moreover, we resolve positively a new special case of the \emph{Generalized Baues Problem}: The Baues poset of all polytopal decompositions of a cyclic polytope of dimension $d \leq 3$ has the homotopy type of a sphere of dimension $n-d-2$. | On Subdivision Posets of Cyclic Polytopes | 13,339 |
A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set $S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational $\alpha>1$, Chow and Long constructed a partition which avoids the numerators of all convergents to $\alpha$, and conjectured that the set $S_\alpha$ which this partition avoided was uniquely avoidable. We prove that the set of numerators of convergents is uniquely avoidable if and only if the continued fraction for $\alpha$ has infinitely many partial quotients equal to 1. We also construct the set $S_\alpha$ and show that it is always uniquely avoidable. | Continued Fractions and Unique Additive Partitions | 13,340 |
We reprove and generalize in a combinatorial way the result of A. Bj\"orner [J.\ Comb.\ Th.\ A {\bf 30}, 1981, pp.~90--100, Theorem 3.3], that order complexes of noncomplemented lattices are contractible, namely by showing that these simplicial complexes are in fact nonevasive, in particular collapsible. | Order complexes of noncomplemented lattices are nonevasive | 13,341 |
We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. In addressing this problem, we study the {\em Shaped Partition Polytope} defined as the convex hull of solutions. The Shaped Partition Problem captures ${\cal N}{\cal P}$-hard problems such as the Max-Cut problem and the Traveling Salesperson problem, and the Shaped Partition Polytope may have exponentially many vertices and facets, even when $k$ or $p$ are fixed. In contrast, we show that when both $k$ and $p$ are fixed, the number of vertices is polynomial in $n$, and all vertices can be enumerated and the optimization problem solved in strongly polynomial time. Explicitly, we show that any Shaped Partition Polytope has $O(n^{k{p\choose 2}})$ vertices which can be enumerated in $O(n^{k^2p^3})$ arithmetic operations, and that any Shaped Partition Problem is solvable in $O(n^{kp^2})$ arithmetic operations. | A Polynomial Time Algorithm for Vertex Enumeration and Optimization over
Shaped Partition Polytopes | 13,342 |
In this paper we study the problem of determining the homology groups of a quotient of a topological space by an action of a group. The method is to represent the original topological space as a homotopy limit of a diagram, and then act with the group on that diagram. Once it is possible to understand what the action of the group on every space in the diagram is, and what it does to the morphisms, we can compute the homology groups of the homotopy limit of this quotient diagram. Our motivating example is the symmetric deleted join of a simplicial complex. It can be represented as a diagram of symmetric deleted products. In the case where the simplicial complex in question is a simplex, we perform the complete computation of the homology groups with $\mathbb Z_p$ coefficients. For the infinite simplex the spaces in the quotient diagram are classifying spaces of various direct products of symmetric groups and diagram morphisms are induced by group homomorphisms. Combining Nakaoka's description of the $\mathbb Z_p$-homology of the symmetric group with a spectral sequence, we reduce the computation to an essentially combinatorial problem, which we then solve using the braid stratification of a sphere. Finally, we give another description of the problem in terms of posets and complete the computation for the case of a finite simplex. | Diagrams of classifying spaces and $k$-fold Boolean algebras | 13,343 |
This note is an extended abstract of my talk at the workshop on Representation Theory and Symmetric Functions, MSRI, April 14, 1997. We discuss the problem of finding an explicit description of the semigroup $LR_r$ of triples of partitions of length $\leq r$ such that the corresponding Littlewood-Richardson coefficient is non-zero. After discussing the history of the problem and previously known results, we suggest a new approach based on the ``polyhedral'' combinatorial expressions for the Littlewood-Richardson coefficients. | Littlewood-Richardson semigroups | 13,344 |
Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev. In this paper we study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We show that the complex of not $2$-connected graphs has the homotopy type of a wedge of $(n-2)!$ spheres of dimension $2n-5$. This answers one of the questions raised by Vassiliev in connection with knot invariants. For this case the $S_n$-action on the homology of the complex is also determined. For complexes of not $2$-connected $k$-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not $(n-2)$-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not $(n-3)$-connected graphs we provide a formula for the generating function of the Euler characteristic. | Complexes of not $i$-connected graphs | 13,345 |
Using earlier results we prove a formula for the number $W_{(n,k)}$ of 2-stack sortable permutations of length $n$ with $k$ runs, or in other words, $k-1$ descents. This formula will yield the suprising fact that there are as many 2-stack sortable permutations with $k-1$ descents as with $k-1$ ascents. We also prove that $W_{(n,k)}$ is unimodal in $k$, for any fixed $n$. | 2-stack sortable permutations with a given number of runs | 13,346 |
A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system $A_{n-1}$. The proof is based on an explicit formula for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases. | Extended Linial Hyperplane Arrangements for Root Systems and a
Conjecture of Postnikov and Stanley | 13,347 |
Connections between $q$-rook polynomials and matrices over finite fields are exploited to derive a new statistic for Garsia and Remmel's $q$-hit polynomial. Both this new statistic $mat$ and another statistic for the $q$-hit polynomial $\xi$ recently introduced by Dworkin are shown to induce different multiset Mahonian permutation statistics for any Ferrers board. In addition, for the triangular boards they are shown to generate different families of Euler-Mahonian statistics. For these boards the $\xi$ family includes Denert's statistic $den$, and gives a new proof of Foata and Zeilberger's Theorem that $(exc,den)$ is jointly distributed with $(des,maj)$. The $mat$ family appears to be new. A proof is also given that the $q$-hit polynomials are symmetric and unimodal. | $q$-Rook polynomials and matrices over finite fields | 13,348 |
The closure of the convex cone generated by all flag $f$-vectors of graded posets is shown to be polyhedral. In particular, we give the facet inequalities to the polar cone of all nonnegative chain-enumeration functionals on this class of posets. These are in one-to-one correspondence with antichains of intervals on the set of ranks and thus are counted by Catalan numbers. Furthermore, we prove that the convolution operation introduced by Kalai assigns extreme rays to pairs of extreme rays in most cases. We describe the strongest possible inequalities for graded posets of rank at most 5. | Linear inequalities for flags in graded posets | 13,349 |
We consider a simplicial complex generaliztion of a result of Billera and Meyers that every nonshellable poset contains the smallest nonshellable poset as an induced subposet. We prove that every nonshellable $2$-dimensional simplicial complex contains a nonshellable induced subcomplex with less than $8$ vertices. We also establish CL-shellability of interval orders and as a consequence obtain a formula for the Betti numbers of any interval order. | Obstructions to Shellability | 13,350 |
We consider a finite set $E$ of points in the $n$-dimensional affine space and two sets of objects that are generated by the set $E$: the system $\Sigma$ of $n$-dimensional simplices with vertices in $E$ and the system $\Gamma$ of chambers. The incidence matrix $A= \parallel a_{\sigma, \gamma}\parallel$, $\sigma \in \Sigma$, $\gamma \in \Gamma$, induces the notion of linear independence among simplices (and among chambers). We present an algorithm of construction of bases of simplices (and bases of chambers). For the case $n=2$ such an algorithm was described in the author's paper {\em Combinatorial bases in systems of simplices and chambers} (Discrete Mathematics 157 (1996) 15--37). However, the case of $n$-dimensional space required a different technique. It is also proved that the constructed bases of simplices are geometrical. | Bases in Systems of Simplices and Chambers | 13,351 |
A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a ``strong'' way, is proved. The result is motivated by a conjecture of Graham's concerning probability correlation inequalities for linear extensions of finite posets. | Chain Decomposition Theorems for Ordered Sets (and Other Musings) | 13,352 |
Jantzen-Seitz partitions are those $p$-regular partitions of~$n$ which label $p$-modular irreducible representations of the symmetric group $S_n$ which remain irreducible when restricted to $S_{n-1}$; they have recently also been found to be important for certain exactly solvable models in statistical mechanics. In this article we study their combinatorial properties via a detailed analysis of their residue symbols; in particular the $p$-cores of Jantzen-Seitz partitions are determined. | Residue symbols and Jantzen-Seitz partitions | 13,353 |
We prove a constant term conjecture of Robbins and Zeilberger (J. Combin. Theory Ser. A 66 (1994), 17-27), by translating the problem into a determinant evaluation problem and evaluating the determinant. This determinant generalizes the determinant that gives the number of all totally symmetric self-complementary plane partitions contained in a $(2n)\times(2n)\times(2n)$ box and that was used by Andrews (J. Combin. Theory Ser. A 66 (1994), 28-39) and Andrews and Burge (Pacific J. Math. 158 (1993), 1-14) to compute this number explicitly. The evaluation of the generalized determinant is independent of Andrews and Burge's computations, and therefore in particular constitutes a new solution to this famous enumeration problem. We also evaluate a related determinant, thus generalizing another determinant identity of Andrews and Burge (loc. cit.). By translating some of our determinant identities into constant term identities, we obtain several new constant term identities. | Determinant identities and a generalization of the number of totally
symmetric self-complementary plane partitions | 13,354 |
We compute the number of all rhombus tilings of a hexagon with sides $a,b+1,c,a+1,b,c+1$, of which the central triangle is removed, provided $a,b,c$ have the same parity. The result is a product of four numbers, each of which counts the number of plane partitions inside a given box. The proof uses nonintersecting lattice paths and a new identity for Schur functions, which is proved by means of the minor summation formula of Ishikawa and Wakayama. A symmetric generalization of this identity is stated as a conjecture. | The number of rhombus tilings of a "punctured" hexagon and the minor
summation formula | 13,355 |
We compute the number of perfect matchings of an $M\times N$ Aztec rectangle where $|N-M|$ vertices have been removed along a line. A particular case solves a problem posed by Propp. Our enumeration results follow from certain identities for Schur functions, which are established by the combinatorics of nonintersecting lattice paths. | Schur function identities and the number of perfect matchings of holey
Aztec rectangles | 13,356 |
Robbins conjectured, and Zeilberger recently proved, that there are 1!4!7!...(3n-2)!/n!/(n+1)!/.../(2n-1)! alternating sign matrices of order n. We give a new proof of this result using an analysis of the six-vertex state model (also called square ice) based on the Yang-Baxter equation. | Another proof of the alternating sign matrix conjecture | 13,357 |
Major Percy A. MacMahon's first paper on plane partitions included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series and Macdonald employing his knowledge of symmetric functions. The purpose of this paper is to simplify Macdonald's proof by providing a direct, inductive proof of his formula which expresses the sum of Schur functions whose partitions fit inside a rectangular box as a ratio of determinants. | Elementary Proof of MacMahon's Conjecture | 13,358 |
Propp conjectured that the number of lozenge tilings of a semiregular hexagon of sides $2n-1$, $2n-1$ and $2n$ which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregular hexagon of sides $a$, $a$ and $b$. We prove explicit formulas for the number of lozenge tilings of these hexagons containing the central unit rhombus, and obtain Propp's conjecture as a corollary of our results. | The number of centered lozenge tilings of a symmetric hexagon | 13,359 |
We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement x_i - x_j = 1, 1 \leq i<j \leq n, is equal to the number of alternating trees on n+1 vertices. Remarkably, these numbers have several additional combinatorial interpretations in terms of binary trees, partially ordered sets, and tournaments. More generally, we give formulae for the number of regions and the Poincar'e polynomial of certain finite subarrangements of the affine Coxeter arrangement of type A_{n-1}. These formulae enable us to prove a "Riemann hypothesis" on the location of zeros of the Poincar'e polynomial. We also consider some generic deformations of Coxeter arrangements of type A_{n-1}. | Deformations of Coxeter hyperplane arrangements | 13,360 |
We show how lattice paths and the reflection principle can be used to give easy proofs of unimodality results. In particular, we give a "one-line" combinatorial proof of the unimodality of the binomial coefficients. Other examples include products of binomial coefficients, polynomials related to the Legendre polynomials, and a result connected to a conjecture of Simion. | Unimodality and the reflection principle | 13,361 |
We give a new proof of Chung and Graham's ``G-descent expansion'' of the classical chromatic polynomial, as well as a special case of the quasi-symmetric function expansion of the path-cycle symmetric function Xi_D. Both proofs rely on Stanley's quasi-symmetric function expansion of the chromatic symmetric function X_G. We also show that Stanley's expansion suggests that a Robinson-Schensted algorithm for (3+1)-free posets---something that has been sought for unsuccessfully for some time---ought to ``respect descents'' in a certain precise sense. | Descents, quasi-symmetric functions, and the chromatic symmetric
function | 13,362 |
Stanley has studied a symmetric function generalization X_G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X_G in terms of elementary symmetric functions has nonnegative coefficients if G is a clawfree incomparability graph. Here we give a combinatorial interpretation of these coefficients by combining Gasharov's work on the conjecture with Egecioglu and Remmel's combinatorial interpretation of the inverse Kostka matrix. This gives a new proof of a partial nonnegativity result of Stanley. As an interesting byproduct we derive a previously unnoticed result relating acyclic orientations to P-tableaux. | A note on a combinatorial interpretation of the e-coefficients of the
chromatic symmetric function | 13,363 |
The Weil Conjectures are applied to the Hessenberg Varieties to obtain interesting information about the combinatorics of descents in the symmetric group. Combining this with elementary linear algebra leads to elegant proofs of some identities from the theory of descents. | Descent identities, Hessenberg varieties, and the Weil conjectures | 13,364 |
We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lov\'asz and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids. | Spectra of semi-regular polytopes | 13,365 |
The cyclic polytope $C(n,d)$ is the convex hull of any $n$ points on the moment curve ${(t,t^2,...,t^d):t \in \reals}$ in $\reals^d$. For $d' >d$, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes $\pi: C(n,d') \to C(n,d)$ which "forgets" the last $d'-d$ coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of $C(n,d)$ which are induced by the map $\pi$. Our main result characterizes the triples $(n,d,d')$ for which the fiber polytope is canonical in either of the following two senses: - all polytopal subdivisions induced by $\pi$ are coherent, - the structure of the fiber polytope does not depend upon the choice of points on the moment curve. We also discuss a new instance with a positive answer to the Generalized Baues Problem, namely that of a projection $\pi:P\to Q$ where $Q$ has only regular subdivisions and $P$ has two more vertices than its dimension. | Fiber polytopes for the projections between cyclic polytopes | 13,366 |
Structure constants for the multiplication of Schubert polynomials by Schur symmetric polynomials are known to be related to the enumeration of chains in a new partial order on S_\infty, which we call the universal k-Bruhat order. Here we present a monoid M for this order and show that $M$ is analogous to the nil-Coxeter monoid for the weak order on S_\infty. For this, we develop a theory of reduced sequences for M. We use these sequences to give a combinatorial description of the structure constants above. We also give combinatorial proofs of some of the symmetry relations satisfied by these structure constants. | A Monoid for the Universal K-Bruhat Order | 13,367 |
We compute the number of rhombus tilings of a hexagon with sides $a+2,b+2,c+2,a+2,b+2,c+2$ with three fixed tiles touching the border. The particular case $a=b=c$ solves a problem posed by Propp. Our result can also be viewed as the enumeration of plane partitions having $a+2$ rows and $b+2$ columns, with largest entry $\le c+2$, with a given number of entries $c+2$ in the first row, a given number of entries 0 in the last column and a given bottom-left entry. | Rhombus Tilings of a Hexagon with Three Fixed Border Tiles | 13,368 |
We find the Martin boundary for the Young-Fibonacci lattice YF. Along with the lattice of Young diagrams, this is the most interesting example of a differential poset. The Martin boundary construction provides an explicit Poisson-type integral representation of non-negative harmonic functions on YF. The latter are in a canonical correspondence with a set of traces on Okada locally semisimple algebra. The set is known to contain all the indecomposable traces. Presumably, all of the traces in the set are indecomposable, though we have no proof of this conjecture. Using a new explicit product formula for Okada characters, we derive precise regularity conditions under which a sequence of characters of finite-dimensional Okada algebras converges to a character of the infinite-dimensional one. | The Martin Boundary of the Young-Fibonacci Lattice | 13,369 |
We study the Young graph with edge multiplicities arising in a Pieri-type formula for Jack symmetric polynomials $P_\mu(x;a)$ with a parameter $a$. Starting with the empty diagram, we define recurrently the `dimensions' $\dim_a$ in the same way as for the Young lattice or Pascal triangle. New proofs are given for two known results. The first is the $a$-hook formula for $\dim_a$, first found by R.Stanley. Secondly, we prove (for all complex $u$ and $v$) a generalization of the identity $\sum\nu(c(b)+u)(c(b)+v)\dim\nu/\dim\mu=(n+1)(n+uv)$, where $\nu$ runs over immediate successors of a Young diagram $\mu$ with $n$ boxes. Here $c(b)$ is the content of a new box $b$. The identity is known to imply the existence of an interesting family of positive definite central functions on the infinite symmetric group. The approach is based on the interpretation of a Young diagram as a pair of interlacing sequences, so that analytic techniques may be used to solve combinatorial problems. We show that when dealing with Jack polynomials $P_\mu(x;a)$, it makes sense to consider `anisotropic' Young diagrams made of rectangular boxes of size $1\times a$. | Anisotropic Young diagrams and Jack symmetric functions | 13,370 |
Let A be a subspace arrangement and let chi(A,t) be the characteristic polynomial of its intersection lattice L(A). We show that if the subspaces in A are taken from L(B_n), where B_n is the type B Weyl arrangement, then chi(A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of chi(A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that our first result deals with all subspace arrangements embedded in B_n while the second deals with all finite Weyl groups but only their hyperplane arrangements. | Characteristic and Ehrhart polynomials | 13,371 |
We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain the M\"obius function in various examples including non-crossing set partitions, shuffle posets, and integer partitions in dominance order. Next we present a generalization of Stanley's theorem that the characteristic polynomial of a semimodular supersolvable lattice factors over the integers. We also give some applications of this second main theorem, including the Tamari lattices. | Mobius functions of lattices | 13,372 |
The generalized Euler number E_{n|k} counts the number of permutations of {1,2,...,n} which have a descent in position m if and only if m is divisible by k. The classical Euler numbers are the special case when k=2. In this paper, we study divisibility properties of a q-analog of E_{n|k}. In particular, we generalize two theorems of Andrews and Gessel about factors of the q-tangent numbers. | Arithmetic properties of generalized Euler numbers | 13,373 |
The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincare polynomial of a finite Coxeter group. | The Wiener polynomial of a graph | 13,374 |
Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux. | The shape of a typical boxed plane partition | 13,375 |
This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems in Enumeration of Matchings: Progress Report" (also available from this server as math.CO/9801061). NOTE: This article has now been superseded by math.CO/9904150. | Twenty Open Problems in Enumeration of Matchings | 13,376 |
This document is a brief summary of progress that has been made on the problems posed in the document "Twenty Open Problems in Enumeration of Matchings" (also available from this server as math.CO/9801060). NOTE: This article has now been superseded by math.CO/9904150. | Twenty Open Problems in Enumeration of Matchings: Progress Report | 13,377 |
This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using ``coupling from the past'' to remove initialization bias from Monte Carlo randomization. The article describes several applications to specific kinds of combinatorial objects such as tilings, constrained lattice paths, and alternating-sign matrices. | Generating Random Elements of Finite Distributive Lattices | 13,378 |
In this paper, we continue the study of domino-tilings of Aztec diamonds. In particular, we look at certain ways of placing ``barriers'' in the Aztec diamond, with the constraint that no domino may cross a barrier. Remarkably, the number of constrained tilings is independent of the placement of the barriers. We do not know of a combinatorial explanation of this fact; our proof uses the Jacobi-Trudi identity. | Domino tilings with barriers | 13,379 |
In this article we study domino tilings of a family of finite regions called Aztec diamonds. Every such tiling determines a partition of the Aztec diamond into five sub-regions; in the four outer sub-regions, every tile lines up with nearby tiles, while in the fifth, central sub-region, differently-oriented tiles co-exist side by side. We show that when n is sufficiently large, the shape of the central sub-region becomes arbitrarily close to a perfect circle of radius n/sqrt(2) for all but a negligible proportion of the tilings. Our proof uses techniques from the theory of interacting particle systems. In particular, we prove and make use of a classification of the stationary behaviors of a totally asymmetric one-dimensional exclusion process in discrete time. | Random Domino Tilings and the Arctic Circle Theorem | 13,380 |
In the theory of two-sided matching markets there are two well-known models: the marriage model (where no money is involved) and the assignment model (where payments are involved). Roth and Sotomayor (1990) asked for an explanation for the similarities in behavior between those two models. We address this question by introducing a common generalization that preserves the two important features: the existence of a stable outcome and the lattice property of the set of stable outcomes. | Stable matching in a common generalization of the marriage and
assignment models | 13,381 |
We show that the well known {\em homotopy complementation formula} of Bj\"orner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, $\widetilde{\mathbf G}_n(R)$ and $\exp_n(X)$ which were introduced and studied by V.~Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets $\exp_n(S^m)$ which leads to a negative answer to a question of Vassilev raised at the workshop ``Geometric Combinatorics'' (MSRI, February 1997). | Combinatorics of Topological Posets:\ Homotopy complementation formulas | 13,382 |
The bandwidth of a graph G is the minimum of the maximum difference between adjacent labels when the vertices have distinct integer labels. We provide a polynomial algorithm to produce an optimal bandwidth labeling for graphs in a special class of block graphs (graphs in which every block is a clique), namely those where deleting the vertices of degree one produces a path of cliques. The result is best possible in various ways. Furthermore, for two classes of graphs that are ``almost'' caterpillars, the bandwidth problem is NP-complete. | Bandwidth and density for block graphs | 13,383 |
For a finite multigraph G, the reliability function of G is the probability R_G(q) that if each edge of G is deleted independantly with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that for any connected multigraph G, if the complex number q is such that R_G(q)=0 then |q|<=1. We verify that this conjectured property of R_G(q) holds if G is a series-parallel network. The proof is by an application of the Hermite-Biehler Theorem and development of a theory of higher-order interlacing for polynomials with only real nonpositive zeros. We conclude by establishing some new inequalities which are satisfied by the f-vector of any matroid without coloops, and by discussing some stronger inequalities which would follow (in the cographic case) from the Brown-Colbourn Conjecture, and are hence true for cographic matroids of series-parallel networks. | Zeros of reliability polynomials and f-vectors of matroids | 13,384 |
We define a contravariant functor K from the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graph X, an abelian group B, and a nonnegative integer j, an element of Hom(K^j(X),B) is a coherent family of B-valued flows on the set of all graphs obtained by contracting some (j-1)-set of edges of X; in particular, Hom(K^1(X),R) is the familiar (real) ``cycle-space'' of X. We show that K(X) is torsion-free and that its Poincare polynomial is the specialization t^{n-k}T_X(1/t,1+t) of the Tutte polynomial of X (here X has n vertices and k components). Functoriality of K induces a functorial coalgebra structure on K(X); dualizing, for any ring B we obtain a functorial B-algebra structure on Hom(K(X),B). When B is commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincare polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows in X, and conclude with ten open problems. | The algebra of flows in graphs | 13,385 |
The concept of coreflexive set is introduced to study the structure of digraphs. New characterizations of line digraphs and nth-order line digraphs are given. Coreflexive sets also lead to another natural way of forming an intersection digraph from a given digraph. | Line digraphs and coreflexive vertex sets | 13,386 |
This paper concerns the enumeration of rotation-type and congruence-type convex polyominoes on the square lattice. These can be defined as orbits of the groups C4, of rotations, and D4, of symmetries of the square acting on (translation- type) polyominoes. In virtue of Burnside's Lemma, it is sufficient to enumerate the various symmetry classes (fixed points) of polyominoes defined by the elements of C4 and D4. Using the Temperley--Bousquet-Melou methodology, we solve this problem and provide explicit or recursive formulas for their generating functions according to width, height and area. We also enumerate the class of asymmetric convex polyominoes, using Moebius inversion, and prove that their number is asymptotically equivalent to the number of convex polyominoes, a fact which is empirically evident. | Enumeration of symmetry classes of convex polyominoes in the square
lattice | 13,387 |
One of the simplest axiomatizations of a matroid is in terms of independent sets. Curiously, no such independent set axiomatization is known for Gelfand and Serganova's WP-matroids (which are ``Coxeter group analogues'' of matroids). Here we state and prove such an axiomatization in the special case of symplectic matroids. No prior knowledge of WP-matroids is assumed; we believe the paper should be accessible and interesting to anyone with some interest in matroids. | An independent set axiomatization for symplectic matroids | 13,388 |
The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to i) the number of polygons, ii) the vertex-color distribution, iii) the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti. | Enumeration of m-ary cacti | 13,389 |
The fixed point property for finite posets of width 3 and 4 is studied in terms of forbidden retracts. The ranked forbidden retracts for width 3 and 4 are determined explicitly. The ranked forbidden retracts for the width 3 case that are linearly indecomposable are examined to see which are minimal automorphic. Part of a problem of Niederle from 1989 is thus solved. | The Fixed Point Property for Posets of Small Width | 13,390 |
By practicing the philosophy of our beloved late master, Marco Schutzenberger, to whose memory this article is dedicated, we give an insightful bijective proof of the three-term recurrence satisfied by the Hipparchus-Schroeder numbers 1,1,3,11,45,197,903, ... | A Classic Proof of a Recurrence for a Very Classical Sequence | 13,391 |
Let G be an n-vertex graph with list-chromatic number $\chi_\ell$. Suppose each vertex of G is assigned a list of t colors. Albertson, Grossman, and Haas conjecture that at least $t n / {\chi_\ell}$ vertices can be colored from these lists. We prove a lower bound for the number of colorable vertices. As a corollary, we show that at least 6/7 of the conjectured number can be colored. | A Lower Bound for Partial List Colorings | 13,392 |
Given a set D of positive integers, the associated distance graph on the integers is the graph with the integers as vertices and an edge between distinct vertices if their difference lies in D. We investigate the chromatic numbers of distance graphs. We show that, if $D = {d_1,d_2,d_3,...}$, with $d_n | d_{n+1}$ for all n, then the distance graph has a proper 4-coloring. We further find the exact chromatic numbers of all such distance graphs. Next, we characterize those distance graphs that have periodic proper colorings and show a relationship between the chromatic number and the existence of periodic proper colorings. | Coloring Distance Graphs on the Integers | 13,393 |
Noam Elkies and Everett Howe independently noticed a certain elegant product formula for the multiple integral \int_R \prod_{1 \le i < j \le k} (x_j-x_i) dx_1 \cdots dx_k, where the region $R$ is the set of $k$-tuples satisfying $0 < x_1 < \cdots < x_k < 1$. Later this formula turned out to be a special case of a formula of Selberg. We prove an apparently different generalization \int_R \det\left(x_i^{a_j-1}\right)dx_1 \cdots dx_k = {\prod_{1 \le i<j \le k}(a_j-a_i)\over \prod_{1 \le i \le k} a_i \prod_{1 \le i<j \le k} (a_j+a_i)}. The key tool is a limiting form of a remarkable identity of Okada for summing the k by k minors of an n by k matrix. | An Application of Okada's Minor Summation Formula | 13,394 |
It is way too soon to teach our computers how to become full-fledged humans. It is even premature to teach them how to become mathematicians, it is even unwise, at present, to teach them how to become combinatorialists. But the time is ripe to teach them how to become experts in a suitably defined and narrowly focused subarea of combinatorics. In this article, I will describe my efforts to teach my beloved computer, Shalosh B. Ekhad, how to be an enumerator of Wilf classes. | Enumeration Schemes And (More Importantly) Their Automatic Generation | 13,395 |
The $OS$ algebra $A$ of a matroid $M$ is a graded algebra related to the Whitney homology of the lattice of flats of $M$. In case $M$ is the underlying matroid of a hyperplane arrangement \A in $\C^r$, $A$ is isomorphic to the cohomology algebra of the complement $\C^r\setminus \bigcup \A.$ Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic $OS$ algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic. We construct, for any given simple matroid $M_0$, a pair of infinite families of matroids $M_n$ and $M'_n$, $n\geq 1$, each containing $M_0$ as a submatroid, in which corresponding pairs have isomorphic $OS$ algebras. If the seed matroid $ M_0$ is connected, then $M_n$ and $M'_n$ have different Tutte polynomials. As a consequence of the construction, we obtain, for any $m$, $m$ different matroids with isomorphic $OS$ algebras. Suppose one is given a pair of central complex hyperplane arrangements $\A_0$ and $\A_1$. Let $\S$ denote the arrangement consisting of the hyperplane $\{0\}$ in $\C^1$. We define the parallel connection $P(\A_0,\A_1)$, an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums $\A_0 \oplus \A_1$ and $\S\oplus P(\A_0,\A_1)$ have diffeomorphic complements. | Orlik-Solomon algebras and Tutte polynomials | 13,396 |
The powerful (and so far under-utilized) Goulden-Jackson Cluster method for finding the generating function for the number of words avoiding, as factors, the members of a prescribed set of `dirty words', is tutorialized and extended in various directions. The authors' Maple implementations, contained in several Maple packages available from this paper's website (http://www.math.temple.edu/~zeilberg/gj.html), are described and explained. | The Goulden-Jackson Cluster Method: Extensions, Applications and
Implementations | 13,397 |
We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg Zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur's identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg Zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument. | A Combinatorial Proof of Bass's Evaluations of the Ihara-Selberg Zeta
Function for Graphs | 13,398 |
Catalan's formula, for the portion of the inheritance that a legitimate child of a 19th-century deceased French gentleman should receive, is given a new proof (using Difference Operators), and generalized. Another, more computationally efficient, formula is also derived. | How Much Should a 19th-Century French Bastard Inherit | 13,399 |
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