Multinomial coefficients modulo a prime
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- by Nikolai A. Volodin PDF
- Proc. Amer. Math. Soc. 127 (1999), 349-353 Request permission
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 \[ q_{l,p}^{(r)}=\min _{p^r\le n<p^{r+1}} G(n,l,p)/ (n+1)^\theta , \] then for any integer $r=1,2,\dots$ \[ 0<q_{l,p}^{(r)}-\liminf _{n\to \infty } G(n,l,p)/n^\theta \le \frac 1{\theta p^r} \left (1+\frac 1{p^r}\right )^{\theta -1}. \]References
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Additional Information
- Nikolai A. Volodin
- Affiliation: The Australian Council for Educational Research, Camberwell 3124, Melbourne, Victoria, Australia
- Email: volodin@acer.edu.au
- Received by editor(s): May 19, 1997
- Communicated by: David E. Rohrlich
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 349-353
- MSC (1991): Primary 11B65, 11B50
- DOI: https://doi.org/10.1090/S0002-9939-99-05079-0
- MathSciNet review: 1628428