A Double Bounded Version of Schur's Partition Theorem

Details A Double Bounded Version of Schur's Partition Theorem A Double Bounded Version of Schur's Partition Theorem

2000-06-27

arxiv.org/abs/math/0006207v2

Mathematics

Combinatorics

20 pages, to appear in Erdos memorial issue of Janos Bolyai Society Definitions in Theorems 2 and 3 corrected, references upda

Scientific paper

Schur's partition theorem states that the number of partitions of n into distinct parts congruent 1, 2 (mod 3) equals the number of partitions of n into parts which differ by >= 3, where the inequality is strict if a part is a multiple of 3. We establish a double bounded refined version of this theorem by imposing one bound on the parts congruent 0,1 (mod 3) and another on the parts congruent 2 (mod 3), and by keeping track of the number of parts in each of the residue classes (mod 3). Despite the long history of Schur's theorem, our result is new, and extends earlier work of Andrews, Alladi-Gordon and Bressoud. We give combinatorial and q-theoretic proofs of our result. The special case L=M leads to a representation of the generating function of the underlying partitions in terms of the q-trinomial coefficients extending a similar previous representation of Andrews.

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