Exact Sprague-Grundy formulas are proven for two new impartial games, PNim on Young diagrams and RNim on hyperrectangles, with a full description of partitions of value one.
The number of nonzero binomial coefficients modulo p^alpha
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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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Nim on Integer Partitions and Hyperrectangles
Exact Sprague-Grundy formulas are proven for two new impartial games, PNim on Young diagrams and RNim on hyperrectangles, with a full description of partitions of value one.