The Zrank Conjecture and Restricted Cauchy Matrices

Details The Zrank Conjecture and Restricted Cauchy Matrices The Zrank Conjecture and Restricted Cauchy Matrices

2005-04-24

arxiv.org/abs/math/0504488v1

Mathematics

Combinatorics

17pages, 6 figures

Scientific paper

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.

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