Alder's conjecture

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Abstract

In 1956, Alder conjectured that the number of partitions of n into parts differing by at least d is greater than or equal to that of partitions of n into parts ≡ ±1 (mod d + 3). The Euler identity, the first Rogers-Ramanujan identity, and a theorem of Schur show that the conjecture is true for d = 1, 2, 3, respectively. In 1971, Andrews proved that the conjecture holds for d = 2r - 1, r ≧ 4. In this paper, we prove the conjecture for all d ≧ 32 and d = 7.

Original languageEnglish (US)
Pages (from-to)67-88
Number of pages22
JournalJournal fur die Reine und Angewandte Mathematik
Issue number616
DOIs
StatePublished - Mar 1 2008

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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