Rook theory. I. Rook equivalence of Ferrers boards

Abstract

We introduce a new tool, the factorial polynomials, to study rook equivalence of Ferrers boards. We provide a set of invariants for rook equivalence as well as a very simple algorithm for deciding rook equivalence of Ferrers boards. We then count the number of Ferrers boards rook equivalent to a given Ferrers board.

Monomial and toric ideals associated to Ferrers graphs

“…Ferrers graphs/tableaux have a prominent place in the literature as they have been studied in relation to chromatic polynomials [2,18], Schubert varieties [16,15], hypergeometric series [29], permutation statistics [9,18], quantum mechanical operators [50], and inverse rook problems [23,16,15,42]. More generally, algebraic and combinatorial aspects of bipartite graphs have been studied in depth (see, e.g., [46,30] and the comprehensive monograph [51]).…”

Rook Poset Equivalence of Ferrers Boards

“…Each initial segment is easily seen to be maximal among all initial segments of rook placements, and hence this element is maximal. The minimal element of (3,3,5,6,6) is [1,2,3,4,5], again trivial to verify.…”

“…For example, consider the Ferrers board given by the partition (3,3,5,6,6). The rook placement [2,1,5,3,4] lies below the rook placement [3,1,5,2,6] in the rook poset: sorting each initial segment, we find that 2 ≤ 3, {1, 2} ≤ {1, 3}, {1, 2, 5} ≤ {1, 3, 5}, {1, 2, 3, 5} ≤ {1, 2, 3, 5}, and {1, 2, 3, 4, 5} ≤ {1, 2, 3, 5, 6}.…”

“…The rook placement [2,1,5,3,4] lies below the rook placement [3,1,5,2,6] in the rook poset: sorting each initial segment, we find that 2 ≤ 3, {1, 2} ≤ {1, 3}, {1, 2, 5} ≤ {1, 3, 5}, {1, 2, 3, 5} ≤ {1, 2, 3, 5}, and {1, 2, 3, 4, 5} ≤ {1, 2, 3, 5, 6}. Some statements about the rook poset are immediate: Proposition 1.5.…”

“…For instance, the maximal element of (3,3,5,6,6) is [3,2,5,6,4]. Each initial segment is easily seen to be maximal among all initial segments of rook placements, and hence this element is maximal.…”