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 \(\{1,2,\ldots,n\}\) 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.