Partial Alternating Sign Matrix Bijections and Dynamics

Abstract

We investigate analogues of alternating sign matrices, called partial alternating sign matrices. We prove bijections between these matrices and several other combinatorial objects. We use an analogue of Wieland's gyration on fully-packed loops, which we relate to the study of toggles and order ideals. Finally, we show that rowmotion on order ideals of a certain poset and gyration on partial fully-packed loop configurations are in equivariant bijection.

Figures (20)

• Figure 1: From left to right, top to bottom: a $4 \times 4$ partial alternating sign matrix, along with its corresponding partial monotone triangle, partial height-function matrix, partial fully-packed loop configuration, rectangular ice configuration, order ideal, and nest of osculating paths.
• Figure 2: A $4 \times 4$ alternating sign matrix along with its matrix of partial column sums and its corresponding monotone triangle.
• Figure 3: The graph $G_{m,n}$.
• Figure 4: The gyration action on a fully-packed loop configuration in $G_{4,4}$, first performing the local action on even squares (shaded on the left) and then odd squares (shaded in the middle). The initial and final fully-packed loops have labeled boundary edges for constructing their link patterns.
• Figure 5: The link patterns for the fully-packed loops from Figure \ref{['fplfig']}.

Theorems & Definitions (52)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: Wieland
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9
• Definition 3.1