Mathematics – Combinatorics
Scientific paper
2012-01-03
Mathematics
Combinatorics
17 pages, 4 figures, 1 table
Scientific paper
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 $(1-t)^{3k-i+1}\sum_{n\geq0}p_{k,i}(n)t^n$ is a polynomial in $t$ with nonnegative integral coefficients and provide combinatorial interpretations of the coefficients by using Stanley's theory of $P$-partitions.
Gessel Ira M.
Lin Zhicong
Zeng Jiang
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