Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers m and n, the partial permutohedron 𝒫(m,n) is the convex hull of all vectors in {0,1,… ,n}^m whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of 𝒫(m,n) , and our methods and results include the following. For any m and n, we obtain a bijection between the nonempty faces of 𝒫(m,n) and certain chains of subsets of {1,… ,m} , thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the h-polynomial of 𝒫(m,n) . For any m and n with n≥ m-1 , we use a pyramidal subdivision of 𝒫(m,n) to establish a recursive formula for the normalized volume of 𝒫(m,n) , from which we then obtain closed expressions for this volume. We also use a sculpting process (in which 𝒫(m,n) is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of 𝒫(m,n) with arbitrary m and fixed n≤ 3 , the normalized volume of 𝒫(m,4) with arbitrary m, and the Ehrhart polynomial of 𝒫(m,n) with fixed m≤ 4 and arbitrary n≥ m-1 .

For positive integers m and n, the partial permutohedron 𝒫(m,n) is a certain integral polytope in ℝ^m, which can be defined as the convex hull of the vectors from {0,1,…,n}^m whose nonzero entries are distinct. For n=m-1, 𝒫(m,m-1) is (after translation by (1,…,1)) the polytope P_m of parking functions of length m, and for n≥ m, 𝒫(m,n) is combinatorially equivalent to an m-stellohedron. The main result of this paper is an explicit expression for the Ehrhart polynomial of 𝒫(m,n) for any m and n with n≥ m-1. The result confirms the validity of a conjecture for this Ehrhart polynomial in arXiv:2207.14253, and the n=m-1 case also answers a question of Stanley regarding the number of integer points in P_m. The proof of the result involves transforming 𝒫(m,n) to a unimodularly equivalent polytope in ℝ^m+1, obtaining a decomposition of this lifted version of 𝒫(m,n) with n≥ m-1 as a Minkowski sum of dilated coordinate simplices, applying a result of Postnikov for the number of integer points in generalized permutohedra of this form, observing that this gives an expression for the Ehrhart polynomial of 𝒫(m,n) with n≥ m-1 as an edge-weighted sum over graphs (with loops and multiple edges permitted) on m labelled vertices in which each connected component contains at most one cycle, and then applying standard techniques for the enumeration of such graphs.

The enumeration of diagonally symmetric alternating sign matrices (DSASMs) is studied, and a Pfaffian formula is obtained for the number of DSASMs of any fixed size, where the entries for the Pfaffian are positive integers given by simple binomial coefficient expressions. This result provides the first known case of an exact enumeration formula for an alternating sign matrix symmetry class in which a simple product formula does not seem to exist. Pfaffian formulae are also obtained for DSASM generating functions associated with several natural statistics, including the number of nonzero strictly upper triangular entries in a DSASM, the number of nonzero diagonal entries in a DSASM, and the column of the unique 1 in the first row of a DSASM. The proofs of these results involve introducing a version of the six-vertex model whose configurations are in bijection with DSASMs of fixed size, and obtaining a Pfaffian expression for its partition function. Various further results and conjectures are also obtained, including some related to the exact enumeration of off-diagonally symmetric alternating sign matrices, and some related to the asymptotic enumeration of DSASMs and other classes of alternating sign matrices.

We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related factorizations involving sums of two Schur polynomials, and certain odd-sized sets of variables. Our results generalize the factorization identities proved by Ciucu and Krattenthaler (2009) [14] for partitions of rectangular shape. We observe that if, in some of the results, the partitions are taken to have rectangular or double-staircase shapes and all of the variables are set to 1, then factorization identities for numbers of certain plane partitions, alternating sign matrices and related combinatorial objects are obtained.

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DAS- ASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang– Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n + 1) × (2n + 1) DASASMs is∏n (3i)! ,andaconjectureofStroganovfrom2008thattheratiobetweenthenumbersof(2n+1)×(2n+1) i=0 (n+i)! DASASMs with central entry −1 and 1 is n/(n + 1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.

A fractional matching of a graph G is a function f from E(G) to the interval [0,1] such that ∑e∈Γ(v)f(e)≤1 for each v∈V(G), where Γ(v) is the set of edges incident to v. The fractional matching number of G, written α∗′(G), is the maximum of ∑e∈E(G)f(e) over all fractional matchings f. For G with n vertices, m edges, positive minimum degree d, and maximum degree D, we prove α∗′(G)≥max{mD,n−md,dD+dn}. For the first two bounds, equality holds if and only if each component of G is r-regular or is bipartite with all vertices in one part having degree r, where r=D for the first bound and r=d for the second. Equality holds in the third bound if and only if G is regular or is (d,D)-biregular.

This paper consists of a review of results for the exact enumeration of alternating sign matrices of fixed size with prescribed values of some or all of the following six statistics: the numbers of generalized inversions and -1's, and the positions of the 1's in the first and last rows and columns. Many of these results are expressed in terms of generating functions.

The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v is a prescribed nonnegative number bv. General theorems which provide conditions for nonemptiness, give a formula for the dimension, and characterize the vertices, edges and face lattices of such polytopes are obtained. Many of these results are expressed in terms of certain spanning subgraphs of G which are associated with subsets or elements of the polytope. For example, it is shown that an element u of the fractional perfect b-matching polytope of G is a vertex of the polytope if and only if each component of the graph of u either is acyclic or else contains exactly one cycle with that cycle having odd length, where the graph of u is defined to be the spanning subgraph of G whose edges are those at which u is positive.

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1's in the first and last rows and columns of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various new identities satisfied by generating functions associated with fewer than six of the statistics. The derivation of the relation involves combining the Desnanot-Jacobi determinant identity with the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions.

It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on nxn alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of -1's in A and the number of 0's to the left of the 1 in the first row of A, and the respective statistics for a DPP D are the number of nonspecial parts in D, the number of special parts in D and the number of n's in D. Here, the result is generalized to include a fourth statistic for each type of object, where this is the number of 0's to the right of the 1 in the last row of an ASM, and the number of (n-1)'s plus the number of rows of length n-1 in a DPP. This generalization is proved using the known equality of the three-statistic generating functions, together with relations which express each four-statistic generating function in terms of its three-statistic counterpart. These relations are obtained by applying the Desnanot-Jacobi identity to determinantal expressions for the generating functions, where the determinants arise from standard methods involving the six-vertex model with domain-wall boundary conditions for ASMs, and nonintersecting lattice paths for DPPs.

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (3) (1983) 340-359] that, for any n, k, m and p, the number of n x n alternating sign matrices (ASMs) for which the 1 of the first row is in column k + 1 and there are exactly m -1's and m + p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASms and DPPs with fixed n as determinants of n x n matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindstrom-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths. C) 2011 Elsevier Inc. All rights reserved.

The combinatorics of certain osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. More specifically, the paths being considered have fixed start and end points on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths are permitted to share lattice points, but not to cross or share lattice edges. Such paths correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they include cases which correspond to alternating sign matrices and various subclasses thereof. Referring to points of the rectangle through which no or two paths pass as vacancies or osculations respectively, the case of primary interest is tuples of paths with a fixed number $l$ of vacancies and osculations. It is then shown that there exist natural bijections which map each such path tuple $P$ to a pair $(t,\eta)$, where $\eta$ is an oscillating tableau of length $l$ (i.e., a sequence of $l+1$ partitions, starting with the empty partition, in which the Young diagrams of successive partitions differ by a single square), and $t$ is a certain, compatible sequence of $l$ weakly increasing positive integers. Furthermore, each vacancy or osculation of $P$ corresponds to a partition in $\eta$ whose Young diagram is obtained from that of its predecessor by respectively the addition or deletion of a square. These bijections lead to enumeration formulae for osculating paths involving sums over oscillating tableaux.

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The case r = 1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension (n-1)(2) whose vertices are the standard alternating sign matrices of size n. It then follows that, for fixed n, these matrices are enumerated by an Ehrhart polynomial in r.

Integrable boundary conditions are studied for critical A–D–E and general graph-based lattice models of statistical mechanics. In particular, using techniques associated with the Temperley–Lieb algebra and fusion, a set of boundary Boltzmann weights which satisfies the boundary Yang–Baxter equation is obtained for each boundary condition. When appropriately specialized, these boundary weights, each of which depends on three spins, decompose into more natural two-spin edge weights. The specialized boundary conditions for the A–D–E cases are naturally in one-to-one correspondence with the conformal boundary conditions of \(\widehat{s\ell }\)(2) unitary minimal conformal field theories. Supported by this and further evidence, we conclude that, in the continuum scaling limit, the integrable boundary conditions provide realizations of the complete set of conformal boundary conditions in the corresponding field theories.

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G . This approach is carried to completion for sl(2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A -D -E classification. We also review the current status for WZW sl(3) theories. Finally, a systematic generalization of the formalism of Cardy–Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk-boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.

The classification of rational conformal field theories is reconsidered from the standpoint of boundary conditions. Solving Cardy's equation expressing the consistency condition on a cylinder is equivalent to finding integer valued representations of the fusion algebra. A complete solution not only yields the admissible boundary conditions but also gives valuable information on the bulk properties.

Reflection equations are used to obtain families of commuting double-row transfer matrices for interaction-round-a-face (IRF) models with fixed and free boundary conditions. We illustrate our methods for the Andrews-Baxter-Forrester (ABF) models which areL-state models associated with the quantum groupU q (su(2)) at a root of unity. We construct elliptic solutions to the reflection equations for the ABF models by a procedure which uses fusion to build the solutions starting from a trivial solution.