Mean Divisibility of Multinomial coefficients

Shigeki Akiyama

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arXiv · 1212.3750

Mean Divisibility of Multinomial coefficients

Shigeki Akiyama

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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.

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Shigeki Akiyama. 2012-12-16. Mean Divisibility of Multinomial coefficients. https://arxiv.org/abs/1212.3750

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