Jacobi-Stirling polynomials and $P$-partitions

Abstract

We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind $ \JS(n+k,n;z)$ by generalizing the analogous results for the Stirling and Legendre-Stirling numbers. More precisely, letting $\JS(n+k,n;z)=p_{k,0}(n)+p_{k,1}(n)z+...+p_{k,k}(n)z^k$, we show that $(1-t{)}^{3k-i+1}{\sum }_{n\ge 0}{p}_{k,i}(n)t^n$ is a polynomial in $t$ with nonnegative integral coefficients and provide combinatorial interpretations of the coefficients by using Stanley's theory of $P$-partitions.

18 pages, 4 figures, 1 table, minor modifications, to appear in European Journal of Combinatorics, 2012

References in corpus (1)

• The Jacobi-Stirling Numbers