Multinomial coefficients modulo a prime

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 $r=1,2,\dots$ \[ 0<q_{l,p}^{(r)}-\liminf _{n\to \infty } G(n,l,p)/n^\theta \le \frac 1{\theta p^r} \left (1+\frac 1{p^r}\right )^{\theta -1}. \]