Real Zeros and the Signless $r$-Associated Stirling Numbers of the First Kind

Abstract

The signless $r$-associated Stirling numbers of the first kind $d_r(n,k)$ counts the number of permutations of the set $\{1,2,\dots ,n\}$ that have exactly $k$ cycles, each of which is of length greater than or equal to $r$, where $r$is a fixed positive integer. F. Brenti obtained that the generating polynomials of the numbers $d_r(n,k)$ have only real zeros. Here we consider the location of zeros of these polynomials.