For more information about this meeting, contact Matthew Katz, James Sellers, George Andrews.

Title: | Still Yet even More Congruence Properties of the Restricted Partition Function p(n,m) |

Seminar: | Combinatorics/Partitions Seminar |

Speaker: | Brandt Kronholm, Juniata College |

Abstract: |

Ramanujan congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. p(n,m) is the restricted partition function that enumerates the number of partitions of n into exactly m parts. The close relationship between p(n) and p(n,m) is clear:
p(n)=p(n,1)+p(n,2)+ ... +p(n,n-1)+p(n,n).
Until recently, the existence of Ramanujan-type congruences was virtually unknown for p(n,m). Let l be an odd prime. In this presentation we will establish explicit Ramanujan-type congruences for p(n,m) modulo any prime l.
In this presentation we will discuss an intriguing extension of a previous result regarding p(n,m), the restricted partition function that enumerates the number of partitions of n into exactly m parts.
This extension reveals further symmetries of the generating function for p(n,m) and may allow us to gain a better understanding of these Ramanujan-like congruences. Moreover, this extension agrees with the Hardy-Ramanujan-Rademacher formula for p(n) when n is negative, namely, p(n)=0.
We may also discuss recent rank and crank statistics computed by our friend Dennis Eichhorn for p(n,m) mod l - though I really don't know too much about them. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 10 / 09 / 2012 |

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