The number of nonzero binomial coefficients modulo p^alpha

Details The number of nonzero binomial coefficients modulo p^alpha The number of nonzero binomial coefficients modulo p^alpha

2010-01-12

arxiv.org/abs/1001.1783v3

Journal of Combinatorics and Number Theory 3 (2011) 15-25

Mathematics

Number Theory

10 pages; publication version

Scientific paper

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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