Wide partitions, Latin tableaux, and Rota's basis conjecture

Abstract

Say that μ is a “subpartition” of an integer partition λ if the multiset of parts of μ is a submultiset of the parts of λ, and define an integer partition λ to be “wide” if for every subpartition μ of λ, μ⩾ μ′ in dominance order (where μ′ denotes the conjugate of μ). Then Brian Taylor and the first author have conjectured that an integer partition λ is wide if and only if there exists a tableau of shape λ such that (1) for all i, the entries in the ith row of the tableau are precisely the integers from 1 to λ i inclusive, and (2) for all j, the entries in the jth column of the tableau are pairwise distinct. This conjecture was originally motivated by Rota's basis conjecture and, if true, yields a new class of integer multiflow problems that satisfy max-flow min-cut and integrality. Wide partitions also yield a class of graphs that satisfy “delta-conjugacy” (in the sense of Greene and Kleitman), and the above conjecture implies that these graphs furthermore have a completely saturated stable set partition. We present several partial results, but the conjecture remains very much open.