Mikael Hansson

The Bruhat order on conjugationinvariant sets of involutions in the symmetric group
dmtcs:2472 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2015,
DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)

https://doi.org/10.46298/dmtcs.2472
The Bruhat order on conjugationinvariant sets of involutions in the symmetric group
Authors: Mikael Hansson ^{1}
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Mikael Hansson
1 Department of Mathematics [Linköping]
Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \subseteq \{0,1,\ldots,n\}$, let \[ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ has $a$ fixed points for some $a \in A$}\}. \] We give a complete characterisation of the sets $A$ for which $F_n^A$, with the order induced by the Bruhat order on $S_n$, is a graded poset. In particular, we prove that $F_n^{\{1\}}$ (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When $F_n^A$ is graded, we give its rank function. We also give a short new proof of the ELshellability of $F_n^{\{0\}}$ (i.e., the set of fixed pointfree involutions), which was recently proved by Can, Cherniavsky, and Twelbeck.