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| Whether a combinatorial sequence is $f$-vector of a simplicial complex is a pop topic in combinatorics and topology. Many combinatorial sequence satisfy this property, such that the Eulerian numbers, the Narayana numbers and so on. Brenti conjectured that the reverse of the sequence of the Stirling number of the seconde kind is $f$-vector of a simplicial complex. Motivated by this problem, we concern with the graphical representation of set partition and construct a class ofsimplicial complexes. Applying it, We prove some combinatorial sequences areHilbert functions, including the positivity of Brenti's conjecture. |
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| Keywords:Applied Mathematics, Stirling number of the second kind, simplicial complex, Hilbert function |
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