Geodesic
Coefficients of Gaussian polynomials modulo \(N\)
Dylan Pentland1
1MIT PRIMES
The electronic journal of combinatorics, Tome 27 (2020) no. 2
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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 $f_{k,R}(n)$.
DOI : 10.37236/7820
Classification : 05A10, 05A15
Mots-clés : Lucas' theorem, quasipolynomial

Dylan Pentland  1

1 MIT PRIMES 
@article{10_37236_7820,
     author = {Dylan Pentland},
     title = {Coefficients of {Gaussian} polynomials modulo {\(N\)}},
     journal = {The electronic journal of combinatorics},
     year = {2020},
     volume = {27},
     number = {2},
     doi = {10.37236/7820},
     zbl = {1444.05008},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/7820/}
}
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Dylan Pentland. Coefficients of Gaussian polynomials modulo \(N\). The electronic journal of combinatorics, Tome 27 (2020) no. 2. doi: 10.37236/7820

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