The zrank conjecture and restricted Cauchy matrices

Guo Guang Yan; Arthur L.B. Yang; Joan J. Zhou

doi:10.1016/j.laa.2005.04.023

The zrank conjecture and restricted Cauchy matrices

Guo Guang Yan, Arthur L.B. Yang, Joan J. Zhou

Epidemiology and Biostatistics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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 x₁ = ⋯ = 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.

Original language	English (US)
Pages (from-to)	371-385
Number of pages	15
Journal	Linear Algebra and Its Applications
Volume	411
Issue number	1-3
DOIs	https://doi.org/10.1016/j.laa.2005.04.023
State	Published - Dec 1 2005

Keywords

Border strip decomposition
Interval sets
Outside decomposition
Rank
Reduced code
Restricted Cauchy matrix
Snakes
Zrank

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2005.04.023

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title = "The zrank conjecture and restricted Cauchy matrices",

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(1t) 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(1t). 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.",

keywords = "Border strip decomposition, Interval sets, Outside decomposition, Rank, Reduced code, Restricted Cauchy matrix, Snakes, Zrank",

author = "Yan, {Guo Guang} and Yang, {Arthur L.B.} and Zhou, {Joan J.}",

note = "Funding Information: This work was done under the auspices of the 973 Project on Mathematical Mechanization, the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China. We thank Professor Richard Stanley for bringing this problem to our attention and for valuable comments. We also thank the referee for the very pertinent comments and suggestions which helped to significantly improve this paper.",

year = "2005",

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doi = "10.1016/j.laa.2005.04.023",

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AU - Yan, Guo Guang

AU - Yang, Arthur L.B.

AU - Zhou, Joan J.

N1 - Funding Information: This work was done under the auspices of the 973 Project on Mathematical Mechanization, the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China. We thank Professor Richard Stanley for bringing this problem to our attention and for valuable comments. We also thank the referee for the very pertinent comments and suggestions which helped to significantly improve this paper.

PY - 2005/12/1

Y1 - 2005/12/1

N2 - 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(1t) 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(1t). 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.

AB - 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(1t) 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(1t). 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.

KW - Border strip decomposition

KW - Interval sets

KW - Outside decomposition

KW - Rank

KW - Reduced code

KW - Restricted Cauchy matrix

KW - Snakes

KW - Zrank

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DO - 10.1016/j.laa.2005.04.023

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VL - 411

SP - 371

EP - 385

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1-3

ER -

The zrank conjecture and restricted Cauchy matrices

Abstract

Keywords

ASJC Scopus subject areas

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