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Coefficients of Gaussian polynomials modulo \(N\). (English) Zbl 1444.05008

Summary: Let \(\left[\begin{smallmatrix} n \\ k \end{smallmatrix}\right]_q\) be a \(q\)-binomial coefficient. R. P. Stanley [Enumerative combinatorics. Vol. 1. With a foreword by Gian-Carlo Rota. Corrected reprint of the 1986 hardback edition. Cambridge: Cambridge University Press (1999; Zbl 0945.05006)] conjectured that the function \(f_{k}(n) = \#\{\alpha : [q^\alpha] \left[\begin{smallmatrix} n \\ k \end{smallmatrix}\right]_q \equiv R \pmod N\}\) is quasi-polynomial for \(N\) prime. We prove this for any integer \(N\) and obtain an expression for the generating function \(F_{k}(x)\) for \(f_{k}(n)\).

MSC:

05A10 Factorials, binomial coefficients, combinatorial functions
05A15 Exact enumeration problems, generating functions

Citations:

Zbl 0945.05006
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References:

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