# Binomial Coefficients, Valuations, and Words

• Eric Rowland
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)

## Abstract

The study of arithmetic properties of binomial coefficients has a rich history. A recurring theme is that p-adic statistics reflect the base-p representations of integers. We discuss many results expressing the number of binomial coefficients $$\left( {\begin{array}{c}n\\ m\end{array}}\right)$$ with a given p-adic valuation in terms of the number of occurrences of a given word in the base-p representation of n, beginning with a result of Glaisher from 1899, up through recent results by Spiegelhofer–Wallner and Rowland.

## Keywords

Binomial coefficients p-adic valuation Regular sequences

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