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Transactions of the American Mathematical Society

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Stanley's zrank conjecture on skew partitions


Authors: William Y. C. Chen and Arthur L. B. Yang
Journal: Trans. Amer. Math. Soc. 360 (2008), 3121-3131
MSC (2000): Primary 05E10, 15A15
DOI: https://doi.org/10.1090/S0002-9947-08-04342-0
Published electronically: January 25, 2008
MathSciNet review: 2379790
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Abstract: We present an affirmative answer to Stanley's zrank conjecture, namely, the zrank and the rank are equal for any skew partition. We show that certain classes of restricted Cauchy matrices are nonsingular and furthermore, the signs are determined by the number of zero entries. We also give a characterization of the rank in terms of the Giambelli-type matrices of the corresponding skew Schur functions. Our approach also applies to the factorial Cauchy matrices and the inverse binomial coefficient matrices.


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Additional Information

William Y. C. Chen
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email: chen@nankai.edu.cn

Arthur L. B. Yang
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email: yang@nankai.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-08-04342-0
Keywords: Zrank, rank, grank, restricted Cauchy matrix, factorial Cauchy matrix, inverse binomial coefficient matrix
Received by editor(s): September 11, 2005
Received by editor(s) in revised form: April 12, 2006
Published electronically: January 25, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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