Abstract:

Schur’s celebrated partition theorem of 1926 asserts the equality D(n) = C(n), where D(n) is the number of partitions of n into parts that differ by ≥ 3 and with no consecutive multiples of 3, and C(n) is the number of partitions on n into distinct parts which are not multiples of 3. In 1989, Basil Gordon and I generalized and refined this theorem by the method of weighted words. In the early 1990s, I mentioned to George Andrews that it would also be worthwhile to investigate another partition function equal to D(n) and C(n), namely, A(n) = the number of partitions of n into odd parts that repeat at most twice. Andrews named the equality D(n) = A(n) as the Alladi-Schur Theorem when he obtained in 2015, the following deep, elegant and surprising refinement of it: Let D(m, n) denote the number of partitions of n of the type enumerated by D(n), with the added condition that the number of parts PLUS the number of even parts is m. Let A(m, n) denote the number of partitions of n into m odd parts that repeat at most twice. Then D(m, n) = A(m, n). Andrews’ proof was by the use of generating function polynomials. In this talk, after surveying major results on Schur’s partition theorem, I will discuss an intricate bijective proof of Andrews’ refinement of the Alladi-Schur Theorem due my PhD student Yazan Alamoudi.