### 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.

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 | |

State | Published - Dec 1 2005 |

Externally published | Yes |

### Fingerprint

### 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

### Cite this

*Linear Algebra and Its Applications*,

*411*(1-3), 371-385. https://doi.org/10.1016/j.laa.2005.04.023

**The zrank conjecture and restricted Cauchy matrices.** / Yan, Guo Guang; Yang, Arthur L B; Zhou, Jin.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 411, no. 1-3, pp. 371-385. https://doi.org/10.1016/j.laa.2005.04.023

}

TY - JOUR

T1 - The zrank conjecture and restricted Cauchy matrices

AU - Yan, Guo Guang

AU - Yang, Arthur L B

AU - Zhou, Jin

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

UR - http://www.scopus.com/inward/record.url?scp=27644585927&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27644585927&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2005.04.023

DO - 10.1016/j.laa.2005.04.023

M3 - Article

AN - SCOPUS:27644585927

VL - 411

SP - 371

EP - 385

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -