Some divisibility properties of binomial and q-binomial coefficients

TL;DR

The paper proves new divisibility properties of binomial and q-binomial coefficients, confirming a conjecture and establishing infinite non-divisibility cases, with implications for q-analogues and related conjectures.

Contribution

It proves a conjecture on divisibility properties of binomial coefficients and introduces new divisibility results, extending to q-binomial coefficients and proposing related conjectures.

Findings

Existence of infinitely many n where binomial coefficients are not divisible by a specific integer.

Divisibility of certain binomial coefficients by linear functions of n.

Positivity and divisibility results for quotients of q-binomial coefficients.

Abstract

We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that $\binom {12n} {3n}$ and $\binom {12n} {4n}$ are divisible by $6n-1$, and that $\binom {330n} {88n}$ is divisible by $66n-1$, for all positive integers $n$. As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of $q$-binomial coefficients by $q$-integers, generalizing the positivity of $q$-Catalan numbers. We also put forward several related conjectures.