Mean Divisibility of Multinomial coefficients

TL;DR

This paper investigates the divisibility properties of sequences formed by multinomial coefficients, establishing conditions under which certain products are rational integers scaled by a fixed constant, based on the greatest common divisor of the parameters.

Contribution

It provides a characterization of when the ratio of products of multinomial coefficients is contained in a scaled integer set, depending on the gcd of the parameters.

Findings

Existence of a positive integer C(k) under gcd condition

Ratio of products is in (1/C(k)) Z for all t

GCD condition is necessary and sufficient

Abstract

Let m_1,...,m_s be positive integers. Consider the sequence defined by multinomial coefficients: a_n=\binom{(m_1+m_2+... +m_s)n}{m_1 n, m_2 n,..., m_s n}. Fix a positive integer k\ge 2. We show that there exists a positive integer C(k) such that \frac{\prod_{n=1}^t a_{kn}}{\prod_{n=1}^t a_n} \in \frac 1{C(k)} \Z for all positive integer t, if and only if GCD(m_1,...,m_s)=1.