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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A $ q$-polynomial identity


Author: Kung-Wei Yang
Journal: Proc. Amer. Math. Soc. 95 (1985), 152-154
MSC: Primary 05A15; Secondary 05A30
MathSciNet review: 796465
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Abstract: We show that the $ q$-polynomial coefficients $ [_{\;r}^{n \cdot m}]$ (see the definition preceding Theorem 1), which are the generating functions of the number of inversions between multisets, satisfy the beautiful identity

$\displaystyle \prod\limits_{i = 1}^n {\left( {\sum\limits_{j = 0}^m {{q^{jm(i -... ... \right) = \sum\limits_{r = 0}^{nm} {[_{\;r}^{n \cdot m}} ]{q^{(_2^r)}}{t^r}} .$


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0796465-8
PII: S 0002-9939(1985)0796465-8
Keywords: $ q$-polynomial identity, $ q$-polynomial coefficient, inversion between multisets
Article copyright: © Copyright 1985 American Mathematical Society