# MacMahon's partial fractions.

## Combinatorics/Partitions Seminar

## Meeting Details

For more information about this meeting, contact Matthew Katz.

**Speaker:** Andrew Sills, Georgia Southern University

**Abstract:** Cayley used ordinary partial fractions decompositions of
1/[(1-x)(1-x^2). . .(1-x^m)] to obtain direct formulas for the number of
partitions of n into at most m parts for several small values of m.
No pattern for general m can be discerned from these, and in particular the
rational coefficients that appear in the partial fraction decomposition
become quite cumbersome for even moderate sized m.
MacMahon gave a decomposition of 1/[(1-x)(1-x^2). . .(1-x^m)] into what
he called "partial fractions of a new and special kind" in which the
coefficients are "easily calculable number[s]" and the sum is indexed by
the partitions of m.
While MacMahon's derived his "new and special" partial fractions using
"combinatory analysis," the aim of this talk is to give a preliminary report
on a fully combinatorial explanation of MacMahon's decomposition. It seems
likely that this will give a combinatorial explanation for the coefficients
that appear in the ordinary partial fraction decompositions, which in turn
can be used to give a formula for the number of partitions of n into
at most m parts for arbitrary m.

## Room Reservation Information

**Room Number:** 106 McAllister

**Date:** 06/09/2015

**Time:** 11:15am - 12:05pm