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An application of Pólya’s enumeration theorem to partitions of subsets of positive integers

Xiao Jun Wu, Chong-Yun Chao (2005)

Czechoslovak Mathematical Journal

Let S be a non-empty subset of positive integers. A partition of a positive integer n into S is a finite nondecreasing sequence of positive integers a 1 , a 2 , ⋯ , a r in S with repetitions allowed such that ∑ i = 1 r a i = n . Here we apply Pólya’s enumeration theorem to find the number ( n ; S ) of partitions of n into S , and the number D P ( n ; S ) of distinct partitions of n into S . We also present recursive formulas for computing ( n ; S ) and D P ( n ; S ) .

An elementary proof of a congruence by Skula and Granville

Romeo Meštrović (2012)

Archivum Mathematicum

Let p ≥ 5 be a prime, and let q p ( 2 ) : = ( 2 p - 1 - 1 ) / p be the Fermat quotient of p to base 2 . The following curious congruence was conjectured by L. Skula and proved by A. Granville q p ( 2 ) 2 ≡ - ∑ k = 1 p - 1 2 k k 2 ( mod p ) . In this note we establish the above congruence by entirely elementary number theory arguments.

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