A local bijection between alternating sign matrices and descending plane partitions and a Striker--Fulmek-type $q$-statistic

Abstract

We prove a local bijection between alternating sign matrices (ASMs) and descending plane partitions (DPPs) in the refined fibres of Behrend--Di Francesco--Zinn-Justin. The map is constructed by moving a boundary line through the ASM and DPP pictures. At each step only a bounded neighbourhood of that boundary is changed. In growth-diagram language, the boundary states are ordinary ASM row frontiers and shifted DPP path frontiers, and the elementary moves are reversible Fomin-type square rules. The proof checks that overlapping local moves are confluent, that every square is invertible, and that the completed boundary again satisfies the ordinary ASM and DPP coordinate inequalities. The only terminal choice is resolved by a radius-two fan rule on the visible height contour. The same local framework gives a statistic on refined ASMs whose generating function is the DPP degree enumerator. The statistic has a Striker--Fulmek quadratic core \[ Q_{\rm SF}(A)= \sum_{1\le r<s\le n}\ \sum_{1\le b<a\le n} (n-r+1)A_{r,a}A_{s,b}, \] plus a bounded fan-boundary correction and, outside the normal boundary gauge, an explicit compact-potential endpoint term. In the normal gauge this is \[ W_{\rm ASM}^{\rm SF}(A)=Q_{\rm SF}(A)+B_{\rm fan}(A). \] Thus the paper gives both a local ASM--DPP bijection and a statistic-preserving refined \(q\)-enumeration. The supplementary files reproduce selected local checks and the displayed arithmetic examples.