arXiv math/03122172003-12-11math.COcond-mat.stat-mechmath-phmath.MP

Proof of two conjectures of Zuber on fully packed loop configurations

F. Caselli, C. Krattenthaler

备注：: 20 pages; AmS-LaTeX
发表信息：: J. Combin. Theory Ser. A 108 (2004), 123-146.

英文摘要：

Two conjectures of Zuber [``On the counting of fully packed loops configurations. Some new conjectures,'' preprint] on the enumeration of configurations in the fully packed loop model on the square grid with periodic boundary conditions, which have a prescribed linkage pattern, are proved. Following an idea of de Gier [``Loops, matchings and alternating-sign matrices,'' Discrete Math., to appear], the proofs are based on bijections between such fully packed loop configurations and rhombus tilings, and the hook-content formula for semistandard tableaux.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

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 $m$ nested arches is polynomial in $m$ if $m$ is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11 (2004), Article #R13].

2005-04-22math-phmath.COmath.MP

Fully packed loop configurations: polynomiality and nested arches

This article proves a conjecture by Zuber about the enumeration of fully packed loops (FPLs). The conjecture states that the number of FPLs whose link pattern consists of two noncrossing matchings which are separated by $m$ nested arches is a polynomial function in $m$ of certain degree and with certain leading coefficient. Contrary to the approach of Caselli, Krattenthaler, Lass and Nadeau (who proved a partial result) we make use of the theory of wheel polynomials developed by Di Francesco, Fonseca and Zinn-Justin. We present a new basis for the vector space of wheel polynomials and a polyno…

2018-03-21math.CO

On the Counting of Fully Packed Loop Configurations. Some new conjectures

New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz, these conjectures are based on the analysis of the ground state of the Temperley-Lieb chain, for periodic boundary conditions and so-called ``identified connectivities'', up to size $2n=22$.

2003-09-25cond-mat.stat-mechmath-phmath.CO

A conjectured formula for Fully Packed Loop configurations in a triangle

We describe a new conjecture involving Fully Packed Loop counting which relates recent observations of Thapper to formulae in the Temperley--Lieb model of loops, and how it implies the Razumov--Stroganov conjecture.

2009-11-24math-phmath.COmath.MP

On some polynomials enumerating Fully Packed Loop configurations

We are interested in the enumeration of Fully Packed Loop configurations on a grid with a given noncrossing matching. By the recently proved Razumov--Stroganov conjecture, these quantities also appear as groundstate components in the Completely Packed Loop model. When considering matchings with p nested arches, these numbers are known to be polynomials in p. In this article, we present several conjectures about these polynomials: in particular, we describe all real roots, certain values of these polynomials, and conjecture that the coefficients are positive. The conjectures, which are of a com…

2010-06-21math-phmath.COmath.MP

On some polynomials enumerating Fully Packed Loops configurations, evaluation at negative values

In this article, we are interested in the enumeration of Fully Packed Loops configurations on a grid with a given noncrossing matching. These quantities also appear as the groundstate components of the Completely Packed Loops model as conjectured by Razumov and Stroganov and recently proved by Cantini and Sportiello. When considering matchings with p nested arches these quantities are known to be polynomials. In a recent article, Fonseca and Nadeau conjectured some unexpected properties of these polynomials, suggesting that these quantities could be combinatorially interpreted even for negativ…

2013-05-24math-phmath.COmath.MP

更多相关论文收起更多论文

Proof of the Razumov-Stroganov conjecture

The Razumov-Stroganov conjecture relates the ground-state coefficients in the even-length dense O(1) loop model to the enumeration of fully-packed loop configuration on the square, with alternating boundary conditions, refined according to the link pattern for the boundary points. Here we prove this conjecture, by mean of purely combinatorial methods. The main ingredient is a generalization of the Wieland proof technique for the dihedral symmetry of these classes, based on the `gyration' operation, whose full strength we will investigate in a companion paper.

2010-03-17cond-mat.stat-mechmath-phmath.CO

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.

2006-07-07math-phmath.COmath.MP

A refined Razumov-Stroganov conjecture

We extend the Razumov-Stroganov conjecture relating the groundstate of the O(1) spin chain to alternating sign matrices, by relating the groundstate of the monodromy matrix of the O(1) model to the so-called refined alternating sign matrices, i.e. with prescribed configuration of their first row, as well as to refined fully-packed loop configurations on a square grid, keeping track both of the loop connectivity and of the configuration of their top row. We also conjecture a direct relation between this groundstate and refined totally symmetric self-complementary plane partitions, namely, in th…

2004-09-08cond-mat.othercond-mat.stat-mechmath-ph

Fully Packed Loop configurations in a triangle

Fully Packed Loop configurations (FPLs) are certain configurations on the square grid, naturally refined according to certain link patterns. If $A_X$ is the number of FPLs with link pattern $X$, the Razumov--Stroganov correspondence provides relations between numbers $A_X$ relative to a given grid size. In another line of research, if $X\cup p$ denotes $X$ with $p$ additional nested arches, then $A_{X\cup p}$ was shown to be polynomial in $p$: the proof gives rise to certain configurations of FPLs in a triangle (TFPLs). In this work we investigate these TFPL configurations and their relation t…

2011-11-25math.CO

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.

2007-11-19math.CO

Meanders: Exact Asymptotics

We conjecture that meanders are governed by the gravitational version of a c=-4 two-dimensional conformal field theory, allowing for exact predictions for the meander configuration exponent α=\sqrt{29}(\sqrt{29}+\sqrt{5})/12, and the semi-meander exponent {\barα}=1+\sqrt{11}(\sqrt{29}+\sqrt{5})/24. This result follows from an interpretation of meanders as pairs of fully packed loops on a random surface, described by two c=-2 free fields. The above values agree with recent numerical estimates. We generalize these results to a score of meandric numbers with various geometries and arbitrary loop …

2000-02-04cond-mat.stat-mechhep-thmath-ph

Combinatorial nature of ground state vector of O(1) loop model

Hanging about a hypothetical connections between the ground state vector for some special spin systems and the alternating-sign matrices, we have found a numerical evidence for the fact that the numbers of the states of the fully packed loop model with fixed link-patterns coincide with the components of the ground state vector of the dense O$(1)$ loop model considered by Batchelor, de Gier and Nienhuis. Our conjecture generalizes in a sense the conjecture of Bosley and Fidkowski, refined by Cohn and Propp, and proved by Wieland.

2001-05-06cond-mathep-thmath-ph

Permutation statistics related to a class of noncommutative symmetric functions and generalizations of the Genocchi numbers

We prove conjectures of the third author [L. Tevlin, Proc. FPSAC'07, Tianjin] on two new bases of noncommutative symmetric functions: the transition matrices from the ribbon basis have nonnegative integral coefficients. This is done by means of two composition-valued statistics on permutations and packed words, which generalize the combinatorics of Genocchi numbers.

2007-10-02math.CO