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The Zrank Conjecture and Restricted Cauchy Matrices

Guo-Guang Yan·Arthur L. B. Yang·Joan J. Zhou
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Abstract

The rank of a skew partition $λ/μ$, denoted $rank(λ/μ)$, is the smallest number $r$ such that $λ/μ$ is a disjoint union of $r$ border strips. Let $s_{λ/μ}(1^t)$ denote the skew Schur function $s_{λ/μ}$ evaluated at $x_1=...=x_t=1, x_i=0$ for $i>t$. The zrank of $λ/μ$, denoted $zrank(λ/μ)$, is the exponent of the largest power of $t$ dividing $s_{λ/μ}(1^t)$. Stanley conjectured that $rank(λ/μ)=zrank(λ/μ)$. 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.

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Guo-Guang Yan, Arthur L. B. Yang, Joan J. Zhou. 2005-04-24. The Zrank Conjecture and Restricted Cauchy Matrices. https://arxiv.org/abs/math/0504488

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