(−1)-enumerations of arrowed Gelfand–Tsetlin patterns

Abstract

Arrowed Gelfand–Tsetlin patterns have recently been introduced to study alternating sign matrices. In this paper, we show that a $(-1)$-enumeration of arrowed Gelfand–Tsetlin patterns can be expressed by a simple product formula. The numbers are up to $2^n$ a one-parameter generalization of the numbers $2^{n(n-1)/2}{\prod }_{j=0}^{n-1}\frac{(4j+2)!}{(n+2j+1)!}$ that appear in recent work of Di Francesco. A second result concerns the $(-1)$-enumeration of arrowed Gelfand–Tsetlin patterns when excluding double-arrows as decoration in which case we also obtain a simple product formula. We are also able to provide signless interpretations of our results. The proofs of the enumeration formulas are based on a recent Littlewood-type identity, which allows us to reduce the problem to the evaluations of two determinants. The evaluations are accomplished by means of the LU-decompositions of the underlying matrices, and an extension of Sister Celine’s algorithm as well as creative telescoping to evaluate certain triple sums. In particular, we use implementations of such algorithms by Koutschan, and by Wegschaider and Riese.