The Zrank Conjecture and Restricted Cauchy Matrices

Abstract

The rank of a skew partition \(\lambda/\mu\), denoted \(rank(\lambda/\mu)\), is the smallest number \(r\) such that \(\lambda/\mu\) is a disjoint union of \(r\) border strips. Let \(s_{\lambda/\mu}(1^t)\) denote the skew Schur function \(s_{\lambda/\mu}\) evaluated at \(x_1=...=x_t=1, x_i=0\) for \(i>t\). The zrank of \(\lambda/\mu\), denoted \(zrank(\lambda/\mu)\), is the exponent of the largest power of \(t\) dividing \(s_{\lambda/\mu}(1^t)\). Stanley conjectured that \(rank(\lambda/\mu)=zrank(\lambda/\mu)\). We show the equivalence between the validity of the zrank conjecture and the nonsingularity of restricted Cauchy matrices. In support of Stanley's conjecture we give affirmative answers for some special cases.