Geodesic
Bruhat order on fixed-point-free involutions in the symmetric group
Matthew Watson1
1North Carolina State University
The electronic journal of combinatorics, Tome 21 (2014) no. 2
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We provide a structural description of Bruhat order on the set $F_{2n}$ of fixed-point-free involutions in the symetric group $S_{2n}$ which yields a combinatorial proof of a combinatorial identity that is an expansion of its rank-generating function. The decomposition is accomplished via a natural poset congruence, which yields a new interpretation and proof of a combinatorial identity that counts the number of rook placements on the Ferrers boards lying under all Dyck paths of a given length $2n$. Additionally, this result extends naturally to prove new combinatorial identities that sum over other Catalan objects: 312-avoiding permutations, plane forests, and binary trees.
DOI : 10.37236/3861
Classification : 05A19
Mots-clés : Bruhat order, fixed-point-free involutions, Dyck paths, rook placements

Matthew Watson  1

1 North Carolina State University 
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Matthew Watson. Bruhat order on fixed-point-free involutions in the symmetric group. The electronic journal of combinatorics, Tome 21 (2014) no. 2. doi: 10.37236/3861

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