The Zrank Conjecture and Restricted Cauchy Matrices
نویسندگان
چکیده
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 sλ/μ(1 ) denote the skew Schur function sλ/μ evaluated at x1 = · · · = xt = 1, xi = 0 for i > t. The zrank of λ/μ, denoted zrank(λ/μ), is the exponent of the largest power of t dividing sλ/μ(1 ). 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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