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 p , we consider the number of ( x , y ) (x, y) with 0 ≤ x , y > p n 0 \leq x, y > p^n for which ( x + y x ) \binom {x+y}{x} is divisible by p z n p^{zn} (but not p z n + 1 p^{zn+1} ) when z n zn is an integer and α > z > β \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 ( ( α , β ) ) p^{n D((\alpha , \beta ))} , where D ( ( α , β ) ) := sup { D ( z ) : α > z > β } D((\alpha , \beta )) := \sup \{ D(z) : \alpha > z > \beta \} and D D is given by an explicit formula. We also develop a “ p p -adic multifractal” theory and show how D 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 q -binomial coefficients of Gauss and the Fibonomial coefficients of Lucas, and finally we extend our results to multinomial coefficients and generalized multinomial coefficients.