Asymptotic prime-power divisibility of binomial, generalized binomial, and multinomial coefficients

Abstract:

This paper presents asymptotic formulas for the abundance of binomial, generalized binomial, multinomial, and generalized multinomial coefficients having any given degree of prime-power divisibility. In the case of binomial coefficients, for a fixed prime $p$, we consider the number of $(x, y)$ with $0 \leq x, y < p^n$ for which $\binom {x+y}{x}$ is divisible by $p^{zn}$ (but not $p^{zn+1}$) when $zn$ is an integer and $\alpha < z < \beta$, say. By means of a classical theorem of Kummer and the probabilistic theory of large deviations, we show that this number is approximately $p^{n D((\alpha , \beta ))}$, where $D((\alpha , \beta )) := \sup \{ D(z) : \alpha < z < \beta \}$ and $D$ is given by an explicit formula. We also develop a “$p$-adic multifractal” theory and show how $D$ may be interpreted as a multifractal spectrum of divisibility dimensions. We then prove that essentially the same results hold for a large class of the generalized binomial coefficients of Knuth and Wilf, including the $q$-binomial coefficients of Gauss and the Fibonomial coefficients of Lucas, and finally we extend our results to multinomial coefficients and generalized multinomial coefficients.