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A q-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets
Li, Yifei1
1 University of Illinois at Springfield Department of Mathematical Sciences and Philosophy One University Plaza MS WUIS 13 Springfield Illinois 62703 (USA)
Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 457-469.

Let f(z)=∑ n=0 ∞ (-1) n z n /n!n!. In their 1975 paper, Carlitz, Scoville and Vaughan provided a combinatorial interpretation of the coefficients in the power series 1/f(z)=∑ n=0 ∞ ω n z n /n!n!. They proved that ω n counts the number of pairs of permutations of the nth symmetric group 𝒮 n with no common ascent. This paper gives a combinatorial interpretation of a natural q-analogue of ω n by studying the top homology of the Segre product of the subspace lattice B n (q) with itself. We also derive an equation that is analogous to a well-known symmetric function identity: ∑ i=0 n (-1) i e i h n-i =0, which then generalizes our q-analogue to a symmetric group representation result.

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DOI: 10.5802/alco.265
Classification: 05E05, 05E10, 05E18, 05E99, 20C30
Keywords: algebraic combinatorics, poset homology, shellability, symmetric functions, symmetric group representation

Li, Yifei 1

1 University of Illinois at Springfield Department of Mathematical Sciences and Philosophy One University Plaza MS WUIS 13 Springfield Illinois 62703 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Li, Yifei. A $q$-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets. Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 457-469. doi : 10.5802/alco.265. https://alco.centre-mersenne.org/articles/10.5802/alco.265/

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