Descents of Permutations in a Ferrers Board

Derangements on a Ferrers board

We study the derangement number on a Ferrers board B = (n × n) − λ with respect to an initial permutation M , that is, the number of permutations on B that share no common points with M . We prove that the derangement number is independent of M if and only if λ is of rectangular shape. We characterize the initial permutations that give the minimal and maximal derangement numbers for a general F...

The Inverse Rook Problem on Ferrers Boards

Rook polynomials have been studied extensively since 1946, principally as a method for enumerating restricted permutations. However, they have also been shown to have many fruitful connections with other areas of mathematics, including graph theory, hypergeometric series, and algebraic geometry. It is known that the rook polynomial of any board can be computed recursively. [19, 18] The naturall...

Invisible permutations and rook placements on a Ferrers board

Rook Poset Equivalence of Ferrers Boards

A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of when two Ding Schubert varieties have iden...

Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions

In a recent paper, Backelin, West and Xin describe a map φ∗ that recursively replaces all occurrences of the pattern k . . . 21 in a permutation σ by occurrences of the pattern (k − 1) . . . 21k. The resulting permutation φ∗(σ ) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map φ∗ commutes with taking the inverse of a permutation. In the BWX paper, the ...