Mathematics – Combinatorics
Scientific paper
2005-04-10
J. Combin. Theory Ser. A 113 (2006), 1061-1071
Mathematics
Combinatorics
12 pages, minor changes, to appear in J. Combin. Theory Ser. A
Scientific paper
10.1016/j.jcta.2005.10.002
Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is log-concave, we prove that the two sequences I_{n,k} and J_{2n,k} are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers a_{n,k} such that $$ \sum_{k=0}^{n-1}I_{n,k}t^k=\sum_{k=0}^{\lfloor (n-1)/2\rfloor}a_{n,k}t^{k}(1+t)^{n-2k-1}. $$ This statement is stronger than the unimodality of I_{n,k} but is also interesting in its own right.
Guo Victor J. W.
Zeng Jiang
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