182 research outputs found
A Bijection between classes of Fully Packed Loops and Plane Partitions
It has recently been observed empirically that the number of FPL
configurations with 3 sets of a, b and c nested arches equals the number of
plane partitions in a box of size a x b x c. In this note, this result is
proved by constructing explicitly the bijection between these FPL and plane
partitions
Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops
We present determinant formulae for the number of tilings of various domains
in relation with Alternating Sign Matrix and Fully Packed Loop enumeration
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
Loops, matchings and alternating-sign matrices
The appearance of numbers enumerating alternating sign matrices in stationary
states of certain stochastic processes is reviewed. New conjectures concerning
nest distribution functions are presented as well as a bijection between
certain classes of alternating sign matrices and lozenge tilings of hexagons
with cut off corners.Comment: LaTeX, 26 pages, 44 figures, extended version of a talk given at the
14th International Conference on Formal Power Series and Algebraic
Combinatorics (Melbourne 2002); Version2: Changed title, expanded some
sections and included more picture
Proof of Razumov-Stroganov conjecture for some infinite families of link patterns
We prove the Razumov--Stroganov conjecture relating ground state of the O(1)
loop model and counting of Fully Packed Loops in the case of certain types of
link patterns. The main focus is on link patterns with three series of nested
arches, for which we use as key ingredient of the proof a generalization of the
Mac Mahon formula for the number of plane partitions which includes three
series of parameters
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
On the number of fully packed loop configurations with a fixed associated matching
We show that the number of fully packed loop configurations corresponding to
a matching with nested arches is polynomial in if is large enough,
thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11
(2004), Article #R13].Comment: AnS-LaTeX, 43 pages; Journal versio
Quantum Knizhnik-Zamolodchikov Equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices
We present multiresidue formulae for partial sums in the basis of link
patterns of the polynomial solution to the level 1 U_q(\hat sl_2) quantum
Knizhnik--Zamolodchikov equation at generic values of the quantum parameter q.
These allow for rewriting and generalizing a recent conjecture [Di Francesco
'06] connecting the above to generating polynomials for weighted Totally
Symmetric Self-Complementary Plane Partitions. We reduce the corresponding
conjectures to a single integral identity, yet to be proved
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