Abstract
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.
Highlights
There is a family of combinatorial identities that express sums over certain Catalan objects in nice closed forms
We provide a structural description of Bruhat order on the set F2n of fixed-pointfree involutions in the symmetric group S2n 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
Summary
There is a family of combinatorial identities that express sums over certain Catalan objects in nice closed forms. In order to dissect the poset F2n, we make use of a bijection φ between fixed-point-free involutions and rook placements on Ferrers boards This bijection is decribed and it is well-known as a part of combinatorial folklore among those concerned with perfect matchings. We exhibit a natural map F2n → Fn whose fibers constitute a poset congruence on the Bruhat order on F2n The quotient modulo this congruence is a natural partial order on Dyck paths. We extend this enumerative result to the context of plane forests and binary trees. Under two natural bijections, the rook number statistic of each Dyck path is equivalent to statistics on these objects This yields additional identities in the family of Catalan sums
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