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Talk: An analog of Euler's theorem on integer partitions (South Alabama)

An analog of Euler's theorem on integer partitions (South Alabama)

Date: 2016/09/15
Occasion: Mathematics Colloquium
Place: University of South Alabama

Abstract

A partition of an integer is a way of writing it as a sum of positive integers (without regard to the order of the summands). It is a famous result of Euler that the number of ways to partition an integer into distinct parts is the same as the number of ways to partition it into odd parts. In this talk, we revisit that theorem (as well as some highlights of the theory of integer partitions) and exhibit a new analog for partitions of fixed perimeter.

This analog arose as a by-product of work on core partitions. A special case of an elegant result due to Anderson proves that the number of (s,s+1)-core partitions is finite and given by the Catalan numbers. Amdeberhan recently conjectured that (s,s+1)-core partitions into distinct parts are enumerated by Fibonacci numbers. We prove this conjecture by enumerating a more general two-parameter family of core partitions into distinct parts.

The talk is intended for a general mathematical audience.

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