Abstract. A plane graph is called symmetric if it is invariant under the reflection across some straight line. We prove a result that expresses the number of perfect matchings of a large class of symmetric graphs in terms of the product of the number of matchings of two subgraphs. When the graph is also centrally symmetric, the two subgraphs are isomorphic and we obtain a counterpart of Jockusch’s squarishness theorem. As applications of our result, we enumerate the perfect matchings of several families of graphs and we obtain new solutions for the enumeration of two of the ten symmetry classes of plane partitions (namely, transposed complementary and cyclically symmetric, transposed complementary) contained in a given box. Finally, we consider symmetry classes of perfect matchings of the Aztec diamond graph and we solve the previously open problem of enumerating the matchings that are invariant under a rotation by 90 degrees. The starting point of this paper is a result [18, Theorem 1] concerning domino tilings of the Aztec diamond compatible with certain barriers. This result has also been generalized and proved bijectively by Propp [17]. We present (see Lemma 1.1) a further generalization,

Mills, Robbins, and Rumsey [8] conjectured, and Zeilberger [13] recently proved, that there 1!4!7!...(3n−2)! are 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. Mills, Robbins, and Rumsey [8] conjectured that: Theorem 1 (Zeilberger) There are n × n alternating sign matrices. A(n) =

The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.

We describe the domino Schensted algorithm of Barbasch, Vogan, Garnkle and van Leeuwen. We place this algorithm in the context of Haiman's mixed and leftright insertion algorithms and extend it to colored words. It follows easily from this description that total color of a colored word maps to the sum of the spins of a pair of 2-ribbon tableaux. Various other properties of this algorithm are described, including an alternative version of the Littlewood-Richardson bijection which yields the q-Littlewood-Richardson coecients of Carre and Leclerc. The case where the ribbon tableau decomposes into a pair of rectangles is worked out in detail. This case is central in recent work [29] on the number of even and odd linear extensions of a product of two chains. 1

An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, −1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries of the square acts in an obvious way on square matrices. For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup. There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P. Stanley suggested the study of those alternating sign matrices in each of these symmetry classes. We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class. Moreover the factorizations of certain of their generating functions point to rather startling connections between several of the symmetry classes and cyclically symmetric plane partitions. 1 1

this paper is to explain in an interesting way some of the combinatorial identities in [Ste1] (labeled collectively therein as the "q = \Gamma1 phenomenon") that arise in connection with the enumeration of symmetry classes of plane partitions. The explanation that we provide here requires a lengthy digression into the representation theory of semisimple Lie groups and their Lie algebras (and this digression will lead us further astray into some interesting questions about conjugacy classes of involutions in such groups), even though the basic reasoning that underpins what we intend to do is quite simple. Because these simple ideas may have broader applicability, and also because there is a danger that the simple ideas may be obscured by the elaborate framework that is constructed around them, we first give a brief indication of the nature of our explanations. We should point out that Greg Kuperberg [Ku] has recently also found explanations of some instances of the q = \Gamma1 phenomenon in symmetry classes of plane partitions. His approach follows the same philosophy but it differs in the details. First let us clarify what is meant by the q = \Gamma1 phenomenon. Suppose that we have a finite collection of combinatorial objects X with an associated generating function F (q). Let us further suppose that there is some "natural" involution x 7! x

. We classify all multiplicity-free products of Schur functions and all multiplicityfree products of characters of SL(n;C). 0. Introduction In this paper, we classify the products of Schur functions that are multiplicity-free; i.e., products for which every coefficient in the resulting Schur function expansion is 0 or 1. We also solve the slightly more general classification problem for Schur functions in any finite number of variables. The latter is equivalent to a classification of all multiplicity-free tensor products of irreducible representations of GL(n) or SL(n). Multiplicity-free representations have many applications, typically based on the fact that their centralizer algebras are commutative, or that their irreducible decompositions are canonical; see the survey article by Howe [H]. We find it surprising that such a natural classification problem seems not to have been considered before. Two well-known examples of multiplicity-free products are the Pieri rules (which cor...

. We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (\Gamma1)-enumeration of these lozenge tilings. In the case that a = b = c, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For m = 0, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants det 0i;jn\Gamma1 \Gamma !ffi ij + \Gamma m+i+j j \Delta\Delta , w...

We present symbolic summation tools in the context of difference fields that help scientists in practical problem solving. Throughout this article we present multi-sum examples which are related to combinatorial problems.

Abstract. Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with “maximal staircases ” removed from some of its vertices. The case of one vertex corresponds to Proctor’s problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose the problem of finding a formula for the number of tilings in this case.