Monotone Triangles and 312 Pattern Avoidance

We introduce a new family 𝒜 n,k of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for k=1, this family is equal to a multivariate generating function involving n+3 variables of objects that extend alternating sign matrices (ASMs), which have recently been introduced by the authors. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. The proof is based on a new antisymmetrizer-to-determinant formula for which we also provide a combinatorial proof, and, although this proof is complicated, it is an important step forward as it is very hard to find combinatorial proofs in this field. In the second part, we relate three specialisations of 𝒜 n,k to weighted enumerations of certain well-known classes of column strict shifted plane partitions that generalise descending plane partitions.

Weak order and descents for monotone triangles

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

Short proof of the ASM theorem avoiding the six-vertex model

Sign-restricted matrices (SRMs) are $(0, \pm 1)$-matrices where, ignoring 0's, the signs in each column alternate beginning with a $+1$ and all partial row sums are nonnegative. The most investigated of these matrices are the alternating sign matrices (ASMs), where the rows also have the alternating sign property, and all row and column sums equal 1. We introduce monotone triangles to represent SRMs and investigate some of their properties and connections to certain polytopes. We also investigate two partial orders for ASMs related to their patterns alternating cycles and show a number of combinatorial properties of these orders.

Michael Somos conjectured a relation between Hankel determinants whose entries ${1\over 2n+1}{3n\choose n}$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of $3$, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.

Totally symmetric self-complementary plane partitions (TSSCPPs) are boxed plane partitions with the maximum possible symmetry. We use the well-known representation of TSSCPPs as a dimer model on a honeycomb graph enclosed in one-twelfth of a hexagon with free boundary to express them as perfect matchings of a family of non-bipartite planar graphs. Our main result is that the edges of the TSSCPPs form a Pfaffian point process, for which we give explicit formulas for the inverse Kasteleyn matrix. Preliminary analysis of these correlations are then used to give a precise conjecture for the limit shape of TSSCPPs in the scaling limit.

A direct bijection between descending plane partitions with no special parts and permutation matrices

We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of Gao–Huang between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.

Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D4, the symmetry group of a square. We examine pattern-avoiding permutations with 180° rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our results involve well-known sequences such as the alternating Fibonacci numbers, triangular numbers, and powers of two.

Permutation patterns and statistics

Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and provide an open-source Python implementation. As a prerequisite, we build up a new theoretical foundation for combinatorial decomposition strategies and combinatorial specifications. We then apply Combinatorial Exploration to the domain of permutation patterns, to great effect. We rederive hundreds of results in the literature in a uniform manner and prove many new ones. These results can be found in a new public database, the Permutation Pattern Avoidance Library (PermPAL) at https://permpal.com. Finally, we give three additional proofs-of-concept, showing examples of how Combinatorial Exploration can prove results in the domains of alternating sign matrices, polyominoes, and set partitions.

A certain subset of the words of length n over the alphabet of non-negative integers satisfying two restrictions has recently been shown to be enumerated by the Catalan number Cn-1. Members of this subset, which we will denote by W(n), have been termed Catalan words or sequences and are closely associated with the 321-avoiding permutations. Here, we consider the problem of enumerating the members of W(n) satisfying various restrictions concerning the containment of certain prescribed subsequences or patterns. Among our results, we show that the generating function counting the members Of W(n) that avoid certain patterns is always rational for four general classes of patterns. Our proofs also provide a general method of computing the generating function for all the patterns in each of the four classes. Closed form expressions in the case of three-letter patterns follow from our general results in several cases. The remaining cases for patterns of length three, which we consider in the final section, may be done by various algebraic and combinatorial methods.

Four Symmetry Classes of Plane Partitions under One Roof

The other dual of MacMahon's theorem on plane partitions