Coefficients of Gaussian Polynomials Modulo $N$
Let $\left[{n \atop k}\right]_q$ be a $q$-binomial coefficient. Stanley conjectured that the function $f_{k,R}(n) = \left|\left\{\alpha : [q^{\alpha}] \left[{n \atop k}\right]_q \equiv R \pmod{N}\right\}\right|$ is quasi-polynomial for $N$ prime. We prove this for any integer $N$ and obtain an expression for the generating function $F_{k,R}(x)$ for $f_{k,R}(n)$.
1985 ◽
Vol 50
(4)
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pp. 791-798
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1964 ◽
Vol 59
(306)
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pp. 487-491
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