More refined enumerations of alternating sign matrices

We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general “d-refined” enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_n,i,j that enumerate such matrices. We give a conjectural explicit formula for A_n,i,j and formulate several other conjectures about the sufficiency of the linear equations to determine the A_n,i,j's and about an extension of the linear equations to the general d-refined enumerations.     

A new proof of the refined alternating sign matrix theorem

In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of n×n alternating sign matrices with a 1 at the top of the ith column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0-1-(−1) matrices generalizing alternating sign matrices.     

Multiply-refined enumeration of alternating sign matrices

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.     

An operator formula for the number of halved monotone triangles with prescribed bottom row

Monotone triangles are certain triangular arrays of integers, which correspond to n×n alternating sign matrices when prescribing (1,2,…,n) as bottom row of the monotone triangle. In this article we define halved monotone triangles, a specialization of which correspond to vertically symmetric alternating sign matrices. We derive an operator formula for the number of halved monotone triangles with prescribed bottom row which is analogous to our operator formula for the number of ordinary monotone triangles [I. Fischer, The number of monotone triangles with prescribed bottom row, Adv. in Appl. Math. 37 (2) (2006) 249-267].     

q-Enumeration of alternating sign matrices with exactly one −1

We enumerate the alternating sign matrices that contain exactly one −1 according to their number of inversions (possibly taking into account the position of the unique non-zero entry in the first row). In conformity with the Mills, Robbins and Rumsey conjectures, this is the same as the enumeration, according to the number of parts, of descending plane partitions with exactly one special part. This is shown by finding a determinantal expression for the generating function of descending plane partitions, transforming it algebraically and extracting recurrences for those with one special part. Finally, we show that the generating function of alternating sign matrices that contain exactly one −1 follows the same recurrences.RésuméOn énumère les matrices à signes alternants qui ne contiennent qu'un seul −1 selon leur nombre d'inversions (en tenant compte possiblement de la position de la seule entrée non nulle de la première ligne). Conformément aux conjectures de Mills, Robbins et Rumsey, ceci revient à l’énumération, selon le nombre de parts, des partitions planes descendantes qui n'ont qu'une seule part spéciale. Pour le démontrer, on obtient d'abord la fonction génératrice des partitions planes descendantes sous forme d'un déterminant qu'on transforme algébriquement pour en extraire des récurrences qui caractérisent celles n'ayant qu'une part spéciale. Finalement, on montre que la fonction génératrice des matrices à signes alternants vérifie les mêmes récurrences.     

New enumeration formulas for alternating sign matrices and square ice partition functions

The refined enumeration of alternating sign matrices (ASMs) of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study, starting with the Refined Alternating Sign Matrix Conjecture of Mills–Robbins–Rumsey (1983) [25], its proof by Zeilberger (1996) [31], and more recent work on doubly-refined and triply-refined enumerations by several authors. In this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the “top–left–bottom” (triply-refined) and “top–left–bottom–right” (quadruply-refined) enumerations. The latter case solves the problem of computing the full boundary correlation function for ASMs. The enumeration formulas are proved by deriving new representations, which are of independent interest, for the partition function of the square ice model with domain wall boundary conditions at the “combinatorial point” η=2π/3$\eta =2\pi /3$.     

Bijective proofs of shifted tableau and alternating sign matrix identities

We give a bijective proof of an identity relating primed shifted gl(n)-standard tableaux to the product of a gl(n) character in the form of a Schur function and . This result generalises a number of well-known results due to Robbins and Rumsey, Chapman, Tokuyama, Okada and Macdonald. An analogous result is then obtained in the case of primed shifted sp(2n)-standard tableaux which are bijectively related to the product of a t-deformed sp(2n) character and . All results are also interpreted in terms of alternating sign matrix (ASM) identities, including a result regarding subsets of ASMs specified by conditions on certain restricted column sums.     

On the weighted enumeration of alternating sign matrices and descending plane partitions

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×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×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 Lindström-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.     

Enumeration of symmetry classes of alternating sign matrices and characters of classical groups

An alternating sign matrix is a square matrix with entries 1, 0 and −1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinant and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round partition functions in terms of irreducible characters of classical groups. In particular, we settle a conjecture on the number of vertically and horizontally symmetric alternating sign matrices (VHSASMs). Dedicated to the memory of David Robbins.     

Alternating sign matrices with one −1 under vertical reflection

We define a bijection that transforms an alternating sign matrix A with one −1 into a pair (N,E) where N is a (so called) neutral alternating sign matrix (with one −1) and E is an integer. The bijection preserves the classical parameters of Mills, Robbins and Rumsey as well as three new parameters (including E). It translates vertical reflection of A into vertical reflection of N. A hidden symmetry allows the interchange of E with one of the remaining two new parameters. A second bijection transforms (N,E) into a configuration of lattice paths called “mixed configuration.”     

On the refined 3-enumeration of alternating sign matrices

An explicit expression for the numbers A(n,r;3) describing the refined 3-enumeration of alternating sign matrices is given. The derivation is based on the recent results of Stroganov for the corresponding generating function. As a result, A(n,r;3)'s are represented as 1-fold sums which can also be written in terms of terminating series of argument 1/4.     

Depth of almost strictly sign regular matrices

The concept of depth of an almost strictly sign regular matrix is introduced and used to simplify some algorithmic characterizations of these matrices.     

A structural characterization of real k-potent matrices

Some well-known characterizations of nonnegative k-potent matrices have been obtained by Flor [P. Flor, On groups of nonnegative matrices, Compositio Math. 21 (1969), pp. 376–382.] and Jeter and Pye [M. Jeter and W. Pye, Nonnegative (s,?t)-potent matrices, Linear Algebra Appl. 45 (1982), pp. 109–121.]. In this article, we obtain a structural characterization of a real k-potent matrix A, provided that (sgn(A)) ^k+1 is unambiguously defined, regardless of whether A is nonnegative or not.     

Linear recurrence relations in the algebra of matrices and applications

We study here some linear recurrence relations in the algebra of square matrices. With the aid of the Cayley–Hamilton Theorem, we derive some explicit formulas for Aⁿ (nr) and e^tA for every r×r matrix A, in terms of the coefficients of its characteristic polynomial and matrices A^j, where 0jr−1.     

Linear maps leaving the alternating group invariant

Let A_n be the group of n×n even permutation matrices, and let V_n be the real linear space spanned by A_n. The purpose of this note is to characterize those linear operators φ on V_n satisfying φ(A_n)=A_n. This answers a question raised by C.K. Li, B.S. Tam, N.K. Tsing [Linear Algebra Appl., to appear].     

The operator formula for monotone triangles - simplified proof and three generalizations

We provide a simplified proof of our operator formula for the number of monotone triangles with prescribed bottom row, which enables us to deduce three generalizations of the formula. One of the generalizations concerns a certain weighted enumeration of monotone triangles which specializes to the weighted enumeration of alternating sign matrices with respect to the number of −1s in the matrix when prescribing (1,2,…,n) as the bottom row of the monotone triangle.     

± sign pattern matrices that allow orthogonality

A sign pattern A is a ± sign pattern if A has no zero entries. A allows orthogonality if there exists a real orthogonal matrix B whose sign pattern equals A. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for ± sign patterns with n − 1 ⩽ N₋(A) ⩽ n + 1 to allow orthogonality.     

Counting permutations by their alternating runs

We find a formula for the number of permutations of [n] that have exactly s runs up and down. The formula is at once terminating, asymptotic, and exact. The asymptotic series is valid for n→∞, uniformly for s?(1−?)n/logn (?>0).     

Alternating mapping functions

We consider functions f from $[n]:=\{1,2,\dots ,n\}$ into itself satisfying that the labels along the iteration orbit of each $i\in [n]$ are forming an alternating sequence, i.e., $i<f(i)>f^2(i)<f^3(i)>?$ or $i>f(i)<f^2(i)>f^3(i)<?$ . We are able to solve the enumeration problem by stating exact and asymptotic formulæ for the number of such so-called alternating n-mapping functions. Furthermore we study the expected component structure of a randomly chosen alternating n-mapping by determining the probability that the underlying mapping graph is connected as well as the limiting distribution of the number of components. Moreover, the corresponding enumeration problem for weakly alternating n-mapping functions has also been solved.