Combinatorics of hexagonal fully packed loop configurations

2014/08/26 by Sabine Beil, Beil, Sabine
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1408.6131

19 pages, 18 figures

arxiv created 2014/08/26 · arxiv updated 2014/08/27

Abstract

In this article, fully packed loop configurations of hexagonal shape (HFPLs) are defined. They generalize triangular fully packed loop configurations. To encode the boundary conditions of an HFPL, a sextuple (l_T,t,r_T;r_B,b,l_B) of 01-words is assigned to it. In the first main result of this article, necessary conditions for the boundary (l_T,t,r_T;r_B,b,l_B) of an HFPL are stated. For instance, the inequality d(r_B)+d(b)+d(l_B)≥ d(l_T)+d(t)+d(r_T)+\vertl_T\vert₁\vertt\vert₀+\vertt\vert₁ \vertr_T\vert₀+\vertr_B\vert₀\vertl_B\vert₁ has to be fulfilled, where \vert⋅\vert_i denotes the number of occurrences of i for i=0,1 and d(⋅) denotes the number of inversions. The other main contribution of this article is the enumeration of HFPLs with boundary (l_T,t,r_T;r_B,b,l_B) such that d(r_B)+d(b)+d(l_B)-d(l_T)-d(t)-d(r_T)-\vertl_T\vert₁\vertt\vert₀- \vertt\vert₁\vertr_T\vert₀-\vertr_B\vert₀\vertl_B\vert₁=0,1. To be more precise, in the first case they are enumerated by Littlewood-Richardson coefficients and in the second case their number is expressed in terms of Littlewood-Richardson coefficients.

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