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arXiv · 1001.1783

The number of nonzero binomial coefficients modulo p^alpha

Eric Rowland
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Abstract

In 1947 Fine obtained an expression for the number of binomial coefficients on row n of Pascal's triangle that are nonzero modulo p. In this paper we use Kummer's theorem to generalize Fine's theorem to prime powers, expressing the number of nonzero binomial coefficients modulo p^alpha as a sum over certain integer partitions. For fixed alpha, this expression can be rewritten to show explicit dependence on the number of occurrences of each subword in the base-p representation of n.

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Eric Rowland. 2011-03-20. The number of nonzero binomial coefficients modulo p^alpha. https://arxiv.org/abs/1001.1783

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