# George Andrews

Affiliation: Pennsylvania State University

Email: andrews@math.psu.edu

Title Of Talk: On the Alladi-Schur Theorem

Abstract: In 1926, I. Schur proved that if $A(n)$ equals the number of partitions of $n$ into parts congruent to 1 or 5 modulo 6, and $B(n)$ equals the number of partitions of n in which any two parts differ by at least 3 and multiples of 3 differ by more than 3, then $A(n)=B(n)$. In the 1990's, K. Alladi noted that if $C(n)$ equals the number of partitions of n into odd parts none repeated more than twice, then also $C(n)=B(n)$. The talk begins with a speculative history on why Schur found $A(n)=B(n)$ but not $B(n)=C(n)$. We then consider the following refinement of the Alladi-Schur theorem:

THEOREM. Let $C(m,n)$ denote the number of partitions among those enumerated by $C(n)$ that have exactly m parts. Let $B(m,n)$ denote the number of partitions among those enumerated by $B(n)$ where the number of odd parts plus twice the number of even parts equals m. The $B(m,n)=C(m,n)$.

We conclude with reasons for believing $B(n)=C(n)$ to be the most natural formulation of Schur's theorem and how this holds implications for future research.