140 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
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
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
Truncated determinants and the refined enumeration of Alternating Sign Matrices and Descending Plane Partitions
Lecture notes for the proceedings of the workshop "Algebraic Combinatorics
related to Young diagram and statistical physics", Aug. 6-10 2012, I.I.A.S.,
Nara, Japan.Comment: 25 pages, 8 figure
Integrable Combinatorics
We review various combinatorial problems with underlying classical or quantum
integrable structures. (Plenary talk given at the International Congress of
Mathematical Physics, Aalborg, Denmark, August 10, 2012.)Comment: 21 pages, 16 figures, proceedings of ICMP1
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 (1983), 340-359]
that, for any n, k, m and p, the number of nxn 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 nxn 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.Comment: v2: published versio
The First Bijective Proof of the Alternating Sign Matrix Theorem Theorem
Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, but a bijective proof for any of these equivalences has been elusive for almost 40 years. In this extended abstract, we provide a sketch of the first bijective proof of the enumeration formula for alternating sign matrices, and of the fact that alternating sign matrices are equinumerous with descending plane partitions. The bijections are based on the operator formula for the number of monotone triangles due to the first author. The starting point for these constructions were known "computational" proofs, but the combinatorial point of view led to several drastic modifications and simplifications. We also provide computer code where all of our constructions have been implemented
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