14 research outputs found
A Natural Bijection between Permutations and a Family of Descending Plane Partitions
We construct a direct natural bijection between descending plane partitions
without any special part and permutations. The directness is in the sense that
the bijection avoids any reference to nonintersecting lattice paths. The
advantage of the bijection is that it provides an interpretation for the
seemingly long list of conditions needed to define descending plane partitions.
Unfortunately, the bijection does not relate the number of parts of the
descending plane partition with the number of inversions of the permutation as
one might have expected from the conjecture of Mills, Robbins and Rumsey,
although there is a simple expression for the number of inversions of a
permutation in terms of the corresponding descending plane partition.Comment: 10 pages, title "modestified", unnecessary definitions removed,
remarks shortene
An update on matters discussed at Oberwolfach by Ian Macdonald in May 1977 and David Robbins in May 1982
Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds
Extended Rate, more GFUN
We present a software package that guesses formulae for sequences of, for
example, rational numbers or rational functions, given the first few terms. We
implement an algorithm due to Bernhard Beckermann and George Labahn, together
with some enhancements to render our package efficient. Thus we extend and
complement Christian Krattenthaler's program Rate, the parts concerned with
guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of
Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's
qGeneratingFunctions.m.Comment: 26 page
Higher Spin Alternating Sign Matrices
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.Comment: 41 pages; v2: minor change
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
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
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
Alternating sign matrix enumeration involving numbers of inversions and -1's, and positions of boundary 1's
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
Higher spin alternating sign matrices
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
Osculating paths and oscillating tableaux
The combinatorics of certain tuples of osculating lattice paths is
studied, and a relationship with oscillating tableaux is obtained.
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 within a tuple are permitted to
share lattice points, but not to cross or share lattice edges. Such
path tuples 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. Of
primary interest here are path tuples with a fixed number l of
vacancies and osculations, where vacancies or osculations are points
of the rectangle through which respectively no or two paths pass. It
is 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 tuples of
osculating paths involving sums over oscillating tableaux