Some combinatorics of rhomboid-shaped fully packed loop configurations

Abstract

The study of rhomboid-shaped fully packed loop configurations (RFPLs) is inspired by the work of Fischer and Nadeau on triangular fully packed loop configurations (TFPLs). By using the same techniques as they did some nice combinatorics for RFPLs arise. To each RFPL and to each oriented RFPL a quadruple of binary words (α ,β ;γ ,δ ) – its so-called boundary – is assigned. There are necessary conditions for the boundary of an RFPL respectively an oriented RFPL. For instance, it has to fulfill the inequality $d(γ )+d(δ )\geq(α )+d(β )+\vert α \vert _0\vert β \vert _1$, where $\vert α \vert _i $ denotes the number of occurrences of $i=0,1$ in α and d(α ) denotes the number of inversions of α . Furthermore, the number of ordinary RFPLs with boundary (α ,β ;γ ,δ ) can be expressed in terms of oriented RFPLs with the same boundary. Finally, oriented RFPLs with boundary (α ,β ;γ ,δ ) such that $d(γ )+d(δ )=d(α )+d(β )+\vert α \vert _0\vert β \vert _1$ are considered. They are in bijection with rhomboid-shaped Knutson-Tao puzzles. Also, Littlewood-Richardson tableaux of defect d are defined. They can be understood as a generalization of Littlewood-Richardson tableaux. Those tableaux are in bijection with rhomboid-shaped Knutson-Tao puzzles.