Total positivity from the exponential Riordan arrays

TL;DR

This paper develops criteria for total positivity in exponential Riordan arrays and applies them to various combinatorial triangles, proving new properties and solving a conjecture related to rook polynomials.

Contribution

It introduces new criteria for total positivity of arrays using exponential Riordan array methods and applies these to prove properties of several combinatorial triangles.

Findings

Established total positivity for Bessel, Lah, and related triangles.

Proved $q$-Stieltjes moment properties and $3$-$q$-log-convexity.

Solved Sokal's conjecture on rook polynomial moment property.

Abstract

Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A 74 (1996)]. Schirmacher [J. Combin. Theory Ser. A 85 (1999)] extended them to $q$-log-concavity and almost $q$-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomial sequence in terms of the exponential formula, the logarithmic formula and the fractional formula, respectively. Finally, we apply our criteria to some triangular arrays satisfying some recurrence relations, including Bessel triangles of two kinds and their generalizations, the Lah triangle and its generalization, the idempotent triangle and some triangles related to binomial coefficients, rook polynomials and Laguerre polynomials. We not only get total positivity of these lower-triangles, and $q$-Stieltjes moment properties and $3$-$q$-log-convexity of their row-generating functions, but also prove that their triangular convolutions preserve Stieltjes moment property. In particular, we solve a conjecture of Sokal on $q$-Stieltjes moment property of rook polynomials.