The Stirling polynomial of a simplicial complex
The author introduces a new encoding of the face numbers of a simplicial complex, its Stirling polynomial, that has a simple expression obtained by multiplying each face number with an appropriate generalized binomial coefficient. It is proved in the article that the face numbers of the barycentric subdivision of the free join of two CW-complexes may be found by multiplying the Stirling polynomials of the barycentric subdivisions of the original complexes. It is also proved that the Stirling polynomial of the order complex of any simplicial poset and of certain graded planar posets has non-negative coefficients. By calculating the Stirling polynomial of the order complex of the \(r\)-cubical lattice of rank \(n+1\) in two ways, the author provides a combinatorial proof for the following identity of Bernoulli polynomials \[ 1+r^{n}\sum_{k=1}^{n}\frac{{n\choose k-1}}{k}\cdot\left (B_{k}\left (x+\frac{1}{r}\right )-B_{k}\left (\frac{1}{r}\right )\right )=(rx+1)^{n}. \] Finally, it is observed that the Stirling polynomials of simplicial complexes associated to the cladistic characters defined by \textit{F. R. McMorris} and \textit{T. Zaslavsky} [Math. Biosci. 54, 3--10 (1981; Zbl 0454.92003)] are equal, up to a shift, to the Stirling polynomials defined by \textit{I. Gessel} and \textit{R. Stanley} [J. Comb. Theory, Ser. A 24, 24--33 (1978; Zbl 0378.05006)].
scientific article