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Gaussian binomial coefficients with negative arguments

Gaussian binomial coefficients with negative arguments
Sam Formichella, Armin Straub — Annals of Combinatorics — Volume 23, Number 3, 2019, Pages 725-748

Abstract

Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas' Theorem on binomial coefficients modulo $$p$$ not only extends naturally to the case of negative entries, but even to the Gaussian case.

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BibTeX

@article{qbinomial-negative-2019,
author = {Sam Formichella and Armin Straub},
title = {Gaussian binomial coefficients with negative arguments},
journal = {Annals of Combinatorics},
year = {2019},
volume = {23},
number = {3},
pages = {725--748},
doi = {10.1007/s00026-019-00472-5},
}