Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness

TL;DR

This paper introduces generalized Stirling cycle and subset triangles, explores their total positivity properties, and proves some conjectures for the case when the parameter r equals 2.

Contribution

It defines new families of matrices generalizing Stirling triangles and investigates their total positivity and real-rootedness properties, proving some conjectures for r=2.

Findings

Proved total positivity properties for r=2.

Established connections between total positivity and real-rootedness.

Formulated conjectures for general r.

Abstract

Given a lower-triangular matrix of real numbers, one can ask the following four total-positivity questions: total positivity of the triangle itself; total positivity of its row-reversal; Toeplitz-total positivity of its row sequences (equivalent to negative-real-rootedness of the row-generating polynomials); and coefficientwise Hankel-total positivity of the sequence of row-generating polynomials. In this paper, we introduce two infinite families of lower-triangular matrices generalising the Stirling cycle and subset triangles, parametrised by an integer $r \ge 1$; we call these the $r$th-order Stirling cycle and subset numbers. We then ask the foregoing four questions for each of these triangles, leading us to several conjectures. We then prove some of these conjectures for the case $r=2$.