2,134 research outputs found
Four symmetry classes of plane partitions under one roof
In previous paper, the author applied the permanent-determinant method of
Kasteleyn and its non-bipartite generalization, the Hafnian-Pfaffian method, to
obtain a determinant or a Pfaffian that enumerates each of the ten symmetry
classes of plane partitions. After a cosmetic generalization of the Kasteleyn
method, we identify the matrices in the four determinantal cases (plain plane
partitions, cyclically symmetric plane partitions, transpose-complement plane
partitions, and the intersection of the last two types) in the representation
theory of sl(2,C). The result is a unified proof of the four enumerations
The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions
We prove that the number of cyclically symmetric, self-complementary plane
partitions contained in a cube of side equals the square of the number of
totally symmetric, self-complementary plane partitions contained in the same
cube, without explicitly evaluating either of these numbers. This appears to be
the first direct proof of this fact. The problem of finding such a proof was
suggested by Stanley
A factorization theorem for lozenge tilings of a hexagon with triangular holes
In this paper we present a combinatorial generalization of the fact that the
number of plane partitions that fit in a box is equal to
the number of such plane partitions that are symmetric, times the number of
such plane partitions for which the transpose is the same as the complement. We
use the equivalent phrasing of this identity in terms of symmetry classes of
lozenge tilings of a hexagon on the triangular lattice. Our generalization
consists of allowing the hexagon have certain symmetrically placed holes along
its horizontal symmetry axis. The special case when there are no holes can be
viewed as a new, simpler proof of the enumeration of symmetric plane
partitions.Comment: 20 page
Symmetry classes of alternating-sign matrices under one roof
In a previous article [math.CO/9712207], we derived the alternating-sign
matrix (ASM) theorem from the Izergin-Korepin determinant for a partition
function for square ice with domain wall boundary. Here we show that the same
argument enumerates three other symmetry classes of alternating-sign matrices:
VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs),
and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was
conjectured by Mills; the others by Robbins [math.CO/0008045]. We introduce
several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (two U-turn
sides), OSASMs (off-diagonally symmetric ASMs), OOSASMs (off-diagonally,
off-antidiagonally symmetric), and UOSASMs (off-diagonally symmetric with
U-turn sides). UASMs generalize VSASMs, while UUASMs generalize VHSASMs
(vertically and horizontally symmetric ASMs) and another new class, VHPASMs
(vertically and horizontally perverse). OSASMs, OOSASMs, and UOSASMs are
related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally
symmetric), DASASMs (diagonally, anti-diagonally symmetric), and TSASMs
(totally symmetric ASMs). We enumerate several of these new classes, and we
provide several 2-enumerations and 3-enumerations.
Our main technical tool is a set of multi-parameter determinant and Pfaffian
formulas generalizing the Izergin-Korepin determinant for ASMs and the Tsuchiya
determinant for UASMs [solv-int/9804010]. We evaluate specializations of the
determinants and Pfaffians using the factor exhaustion method.Comment: 16 pages, 16 inline figures. Introduction rewritten with more
motivation and context. To appear in the Annals of Mathematic
Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions
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 (Advances in combinatorial
mathematics, 39-59, 2009) 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.Comment: 22 pages; v2: minor changes, published versio
Exact conjectured expressions for correlations in the dense O loop model on cylinders
We present conjectured exact expressions for two types of correlations in the
dense O loop model on square lattices with periodic
boundary conditions. These are the probability that a point is surrounded by
loops and the probability that consecutive points on a row are on the
same or on different loops. The dense O loop model is equivalent to the
bond percolation model at the critical point. The former probability can be
interpreted in terms of the bond percolation problem as giving the probability
that a vertex is on a cluster that is surrounded by \floor{m/2} clusters and
\floor{(m+1)/2} dual clusters. The conjectured expression for this
probability involves a binomial determinant that is known to give weighted
enumerations of cyclically symmetric plane partitions and also of certain types
of families of nonintersecting lattice paths. By applying Coulomb gas methods
to the dense O loop model, we obtain new conjectures for the asymptotics
of this binomial determinant.Comment: 17 pages, replaced by version accepted by JSTA
On the link pattern distribution of quarter-turn symmetric FPL configurations
We present new conjectures on the distribution of link patterns for
fully-packed loop (FPL) configurations that are invariant, or almost invariant,
under a quarter turn rotation, extending previous conjectures of Razumov and
Stroganov and of de Gier. We prove a special case, showing that the link
pattern that is conjectured to be the rarest does have the prescribed
probability. As a byproduct, we get a formula for the enumeration of a new
class of quasi-symmetry of plane partitions.Comment: 12 pages, 6 figures. Submitted to FPSAC 200
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