Ferrers Graphs and Related Ideals

This abstract is essentially taken from the introduction of the paper Monomial and toric ideals associated to Ferrers graphs [13], written jointly with Alberto Corso. A Ferrers graph is a bipartite graph on two distinct vertex sets X = {x1, . . . , xn} and Y = {y1, . . . , ym} such that if (xi, yj) is an edge of G, then so is (xp, yq) for 1 ≤ p ≤ i and 1 ≤ q ≤ j. In addition, (x1, ym) and (xn, y1) are required to be edges of G. For any Ferrers graph G there is an associated sequence of non-negative integers λ = (λ1, λ2, . . . , λn), where λi is the degree of the vertex xi. Notice that the defining properties of a Ferrers graph imply that λ1 = m ≥ λ2 ≥ · · · ≥ λn ≥ 1; thus λ is a partition. Alternatively, we can associate to a Ferrers graph a diagram Tλ, dubbed Ferrers tableau, consisting of an array of n rows of cells with λi adjacent cells, left justified, in the i-th row. Ferrers graphs/tableaux have a prominent place in the literature as they have been studied in relation to chromatic polynomials [2, 20], Schubert varieties [18, 17], hypergeometric series [31], permutation statistics [9, 20], quantum mechanical operators [51], inverse rook problems [25, 18, 17, 44]. More generally, algebraic and combinatorial aspects of bipartite graphs have been studied in depth (see, e.g., [47, 32] and the comprehensive monograph [52]). In this paper, which is the first of a series [14, 15], we are interested in the algebraic properties of the edge ideal I = I(G) and the toric ring K[G] associated to a Ferrers graph G. The edge ideal is the monomial ideal of the polynomial ring R = K[x1, . . . , xn, y1, . . . , ym] over the field K that is generated by the monomials of the form xiyj, whenever the pair (xi, yj) is an edge of G. K[G] is instead the monomial subalgebra generated by the elements xiyj. An example is illustrated in Figure 1: