(1)Enumerations of arrowed GelfandTsetlin patterns
Abstract
Arrowed GelfandTsetlin patterns have recently been introduced to study alternating sign matrices. In this paper, we show that a $(1)$enumeration of arrowed GelfandTsetlin patterns can be expressed by a simple product formula. The numbers are a oneparameter generalization of the numbers $2^{n(n1)/2} \prod_{j=0}^{n1} \frac{(4j+2)!}{(n+2j+1)!}$ that appear in recent work of Di Francesco. A second result concerns the (1)enumeration of arrowed GelfandTsetlin patterns when excluding doublearrows 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 Littlewoodtype identity, which allows us to reduce the problem to the evaluations of two determinants. The evaluations are accomplished by means of the LUdecompositions 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.04164
 arXiv:
 arXiv:2302.04164
 Bibcode:
 2023arXiv230204164F
 Keywords:

 Mathematics  Combinatorics;
 05A05;
 05A15;
 05A19;
 15B35;
 82B20;
 82B23
 EPrint:
 21 pages, accepted manuscript for "European Journal of Combinatorics"