7,169 research outputs found
A doubly-refined enumeration of alternating sign matrices and descending plane partitions
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.Comment: 28 pages; v2: published versio
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, its proof by Zeilberger, and more
recent work on doubly-refined and triply-refined enumeration 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" 2{\pi}/3.Comment: 35 page
Vertically symmetric alternating sign matrices and a multivariate Laurent polynomial identity
In 2007, the first author gave an alternative proof of the refined
alternating sign matrix theorem by introducing a linear equation system that
determines the refined ASM numbers uniquely. Computer experiments suggest that
the numbers appearing in a conjecture concerning the number of vertically
symmetric alternating sign matrices with respect to the position of the first 1
in the second row of the matrix establish the solution of a linear equation
system similar to the one for the ordinary refined ASM numbers. In this paper
we show how our attempt to prove this fact naturally leads to a more general
conjectural multivariate Laurent polynomial identity. Remarkably, in contrast
to the ordinary refined ASM numbers, we need to extend the combinatorial
interpretation of the numbers to parameters which are not contained in the
combinatorial admissible domain. Some partial results towards proving the
conjectured multivariate Laurent polynomial identity and additional motivation
why to study it are presented as well
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.Comment: 62 pages; v3 slightly updated relative to published versio
Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order
For each , we count diagonally and antidiagonally
symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal
number of 's along the diagonal and the antidiagonal, as well as
DASASMs of fixed odd order with a minimal number of 's along the diagonal
and the antidiagonal. In these enumerations, we encounter product formulas that
have previously appeared in plane partition or alternating sign matrix
counting, namely for the number of all alternating sign matrices, the number of
cyclically symmetric plane partitions in a given box, and the number of
vertically and horizontally symmetric ASMs. We also prove several refinements.
For instance, in the case of DASASMs with a maximal number of 's along the
diagonal and the antidiagonal, these considerations lead naturally to the
definition of alternating sign triangles. These are new objects that are
equinumerous with ASMs, and we are able to prove a two parameter refinement of
this fact, involving the number of 's and the inversion number on the ASM
side. To prove our results, we extend techniques to deal with triangular
six-vertex configurations that have recently successfully been applied to
settle Robbins' conjecture on the number of all DASASMs of odd order.
Importantly, we use a general solution of the reflection equation to prove the
symmetry of the partition function in the spectral parameters. In all of our
cases, we derive determinant or Pfaffian formulas for the partition functions,
which we then specialize in order to obtain the product formulas for the
various classes of extreme odd DASASMs under consideration.Comment: 41 pages, several minor improvements in response to referee's
comments. Final version. Matches published version except for very minor
change
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