Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. Sun
The electronic journal of combinatorics, Tome 21 (2014) no. 2
For all positive integers $n$, we prove the following divisibility properties:\[ (2n+3){2n\choose n} \left|3{6n\choose 3n}{3n\choose n},\right. \quad\text{and}\quad(10n+3){3n\choose n} \left|21{15n\choose 5n}{5n\choose n}.\right. \]This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers $m$ and $n$, the product $am{am+bm-1\choose am}{an+bn\choose an}$ is divisible by $m+n$. In fact, the latter result can be further generalized to the $q$-binomial coefficients and $q$-integers case, which generalizes the positivity of $q$-Catalan numbers. We also propose several related conjectures.
DOI :
10.37236/4258
Classification :
11B65, 05A10, 05A30
Mots-clés : binomial coefficients, congruences, \(p\)-adic order, \(q\)-Catalan numbers, reciprocal and unimodal polynomials
Mots-clés : binomial coefficients, congruences, \(p\)-adic order, \(q\)-Catalan numbers, reciprocal and unimodal polynomials
Affiliations des auteurs :
Victor J. W. Guo 1
@article{10_37236_4258,
author = {Victor J. W. Guo},
title = {Proof of two divisibility properties of binomial coefficients conjectured by {Z.-W.} {Sun}},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {2},
doi = {10.37236/4258},
zbl = {1305.11013},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4258/}
}
Victor J. W. Guo. Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. Sun. The electronic journal of combinatorics, Tome 21 (2014) no. 2. doi: 10.37236/4258
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