Orthogonal polynomials and Smith normal form
Combinatorics
2017-04-13 v1
Abstract
Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for q-Catalan numbers, q-Motzkin numbers, q-Schr\"oder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q-double factorials, as well as generating functions for permutations with eight statistics.
Cite
@article{arxiv.1704.03539,
title = {Orthogonal polynomials and Smith normal form},
author = {Alexander R. Miller and Dennis Stanton},
journal= {arXiv preprint arXiv:1704.03539},
year = {2017}
}