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Number of multinomial coefficients not divisible by a prime. (English) Zbl 0816.11011

For \(n,\ell\in \mathbb{N}\) fixed, a prime \(p\) and \(N\geq 1\) let \(g(n, \ell, p^ N)\) be the number of multinomial coefficients \({{n!} \over {\dot x_ 1! \cdots \dot x_ \ell!}}\) (\(\dot x_ i\geq 0\), \(i=1,\dots, \ell\); \(\dot x_ 1+ \dots+ \dot x_ \ell=n\)) which are not divisible by \(p^ N\) and \(G(n, \ell, p^ N):= \sum_{k=1}^{n-1} g(n, \ell, p^ N)\). Extending investigations of Harborth in the case \(\ell=2\), \(p^ N=2\) it is shown first that \(\limsup_{n\to\infty}\) \(G(n,\ell, p)/ n^ \theta=1\) \((\theta:= \log_ p (\ell, p-1))\), and further that if \(q_ r:= G(n_ r, \ell, 2)/ n_ r^ \theta\) then \((q_ n )_{n\geq 1}\) is strictly decreasing if \(n_ 0=1\) and \(n_{r+1}= 2n_ r \pm 1\) where \(+\) or \(-\) is chosen such that \(q_{r+1}\) becomes minimal.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11B50 Sequences (mod \(m\))
05A10 Factorials, binomial coefficients, combinatorial functions
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