Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Multinomial coefficients modulo a prime

Author(s): 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 and power $n=j_1+\cdots+j_l$ . Let be the number of m.c. that are not divisible by and have order with powers which are not larger than . 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 $r=1,2,\dots$

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

References:

1.: L. Carlitz (1967), The number of binomial coefficients divisible by a fixed power of a prime, Rend Circ. Mat. Palermo 16, 299-320. MR 40:2554
2.: L. Carlitz (1970), Distribution of binomial coefficients, Riv. Mat. Univ. Parma 11, 45-64. MR 47:45
3.: H. Harborth (1977), Number of odd binomial coefficients, Proc. Amer. Math. Soc. 62, 19-22. MR 55:2725
4.: F. T. Howard (1971), The number of binomial coefficients divisible by a fixed power of 2, Proc. Amer. Math. Soc. 29, 236-242. MR 46:1603
5.: F. T. Howard (1973), The number of binomial coefficients divisible by a fixed power of a prime, Proc. Amer. Math. Soc. 37, 358-362. MR 46:8842
6.: F. T. Howard (1974), The number of multinomial coefficients divisible by a fixed power of a prime, Pacific J. Math. 50, 99-108. MR 49:2513
7.: F. T. Howard (1993), Multinomial and $Q$ -binomial coefficients modulo 4 and modulo $p$ , The Fibonacci Quarterly 31, 53-64. MR 94c:11019
8.: R. J. Martin and G. L. Mullen (1984), Reducing multinomial coefficients modulo a prime power, Computers and Mathematics with Applications 10, 37-41. MR 85a:05007
9.: A. H. Stein (1989), Binomial coefficients not divisible by a prime, Number Theory (New York, 1985/1988), Lecture Notes in Mathematics, vol. 1383, Berlin-New York, Springer, 170-177. MR 91c:11012
10.: K. B. Stolarsky (1975), Digital sums and binomial coefficients, Notices Amer. Math. Soc. 22, A-669.
11.: K. B. Stolarsky (1977), Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math. 32, 717-730. MR 55:12621
12.: N. A. Volodin (1989), Distribution $\operatorname{mod}p^N$ of polynomial coefficients, (Russian) Mat. Zametki 45 (1989), 24-29. MR 90i:11022
13.: N. A. Volodin (1994), Number of multinomial coefficients not divisible by a prime, The Fibonacci Quarterly 32, 402-406. MR 95j:11017
14.: B. Wilson (1996a), Asymptotic behavior of Pascal's triangle modulo a prime, Acta Arith. 83 (1998), 105-116. CMP 98:06
15.: B. Wilson (1996b), LIM INF bounds for multinomial coefficients modulo a prime, Preprint, SUNY College at Brockport, New York, 16 p.

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