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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Multinomial coefficients modulo a prime

Author: Nikolai A. Volodin
Journal: Proc. Amer. Math. Soc. 127 (1999), 349-353
MSC (1991): Primary 11B65, 11B50
MathSciNet review: 1628428
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Abstract: We say that the multinomial coefficient (m.c.) $(j_1,\dots, j_l)=n!/ (j_1!\cdots j_l!)$ has order $l$ and power $n=j_1+\cdots+j_l$. Let $G(n,l,p)$ be the number of m.c. that are not divisible by $p$ and have order $l$ with powers which are not larger than $n$. If $\theta =\log _p(l,p-1)$ and

\begin{displaymath}q_{l,p}^{(r)}=\min _{p^r\le n<p^{r+1}} G(n,l,p)/ (n+1)^\theta, \end{displaymath}

then for any integer $r=1,2,\dots$

\begin{displaymath}0<q_{l,p}^{(r)}-\liminf _{n\to\infty} G(n,l,p)/n^\theta\le\frac 1{\theta p^r} \left(1+\frac 1{p^r}\right)^{\theta-1}. \end{displaymath}

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Nikolai A. Volodin
Affiliation: The Australian Council for Educational Research, Camberwell 3124, Melbourne, Victoria, Australia

Received by editor(s): May 19, 1997
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society