On Schur's second partition theorem

Abstract

In 1926, I. J. Schur proved the following theorem on partitions [3].The number of partitions of n into parts congruent to ±1 (mod 6) is equal to the number of partitions of n of the form ₁ + …+b_s = n, where b_i–b_i+1 ≧ 3 and, if 3 ∣ b_i, then b_i–_bi+1 > 3.Schur's proof was based on a lemma concerning recurrence relations for certain polynomials. In 1928, W. Gleissberg gave an arithmetic proof of a strengthened form of Schur's theorem [2]; however, the combinatorial reasoning in Gleissberg's paper becomes very intricate.