Jacobi-Stirling polynomials and P-partitions

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)^3^k^-^i^+^1@?"n">="0p"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.

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