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- W2949232015 abstract "Pak and Panova recently proved that the $q$-binomial coefficient ${m+n choose m}_q$ is a strictly unimodal polynomial in $q$ for $m,n geq 8$, via the representation theory of the symmetric group. We give a direct combinatorial proof of their result by characterizing when a product of chains is strictly unimodal and then applying O'Hara's structure theorem for the partition lattice $L(m,n)$. In fact, we prove a stronger result: if $m, n geq 8d$, and $2d leq r leq mn/2$, then the $r$-th rank of $L(m,n)$ has at least $d$ more elements that the next lower rank." @default.
- W2949232015 created "2019-06-27" @default.
- W2949232015 creator A5027846781 @default.
- W2949232015 date "2014-02-05" @default.
- W2949232015 modified "2023-09-28" @default.
- W2949232015 title "A combinatorial proof of strict unimodality for $q$-binomial coefficients" @default.
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