Stability of combinatorial polynomials and its applications

Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to make a systematical study on the stability of polynomials in combinatorics. Applying the characterizations of Borcea and Brändén concerning linear operators preserving stability, we present criteria for real stability and Hurwitz stability of recursive polynomials. We also give a criterion for Hurwitz stability of the Turán expressions of recursive polynomials. As applications of these criteria, we derive some stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating runs polynomials of types A and B and solve a conjecture concerning Hurwitz stability of alternating runs polynomials defined on a dual set of Stirling permutations. Furthermore, we prove that the Hurwitz stability of any symmetric polynomial implies its semi-γ-positivity. We study a class of symmetric polynomials and derive many nice properties including Hurwitz stability, semi-γ-positivity, non γ-positivity, unimodality, strong q-log-convexity, the Jacobi continued fraction expansion and the relation with derivative polynomials. In particular, these properties of the alternating descents polynomials of types A and B can be obtained in a unified approach. Finally, based on the h-polynomials from combinatorial geometry, we use real stability to prove a criterion for zeros interlacing between a polynomial and its reciprocal polynomial, which in particular implies the alternatingly increasing property of the original polynomial. This criterion extends a result of Brändén and Solus and unifies such properties for many combinatorial polynomials, including ascent polynomials for k-ary words, descent polynomials on signed Stirling permutations and colored permutations and q-analog of descent polynomials on colored permutations, and so on. Furthermore, we also obtain a recurrence relation and zeros interlacing of q-analog of descent polynomials on colored permutations that extend some results of Brändén and Brenti. In addition, as an application of Hurwitz stability, we prove the alternatingly increasing property and zeros interlacing for two kinds of peak polynomials on the dual set of Stirling permutations. ∗Supported partially by the National Natural Science Foundation of China (Nos. 11971206, 12022105) and the Natural Science Fund for Distinguished Young Scholars of Jiangsu Province (No. BK20200048). Email address: ding-mj@hotmail.com (M.-J. Ding), bxzhu@jsnu.edu.cn (B.-X. Zhu)

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