Binomial predictors

TL;DR

This paper characterizes binomial predictors in base p by relating the set of binomial coefficients divisible by p^k to the base p expansion of n+1, providing a complete classification of such predictors.

Contribution

It offers a full description and characterization of binomial predictors in base p, connecting divisibility properties of binomial coefficients to base p expansions.

Findings

Identifies the structure of binomial predictors in base p.

Provides explicit formulas linking binomial coefficient divisibility to base p digits.

Completes the classification of binomial predictors in the given setting.

Abstract

For a prime p and nonnegative integers n,k, consider the set A_{n,k}^{(p)}={x is in [0,1,...,n]: p^k||binom {n} {x}}. Let the expansion of n+1 in base p be: n+1=alpha_{0} p^{\nu}+alpha_{1}p^{nu-1}+...+alpha_{nu}, where 0<=alpha_{i}<= p-1,i=0,...,nu. Then the number n is called a binomial predictor in base p,if |A_{n,k}^{(p)}|=alpha_{k}p^{nu-k},k=0,1,...,nu. We give a full description of the binomial predictors in base p.