DATES:
- Part 1 - Tuesday, September 14 (2010), at 1:55pm
- Part 2 - Tuesday, September 21 (2010), at 1:55pm
- Part 3 - Tuesday, September 28 (2010), at 1:55pm
PLACE: LIT 368
SPEAKER:
Frank Garvan
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TITLE:
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Higher Order SPT-Functions
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ABSTRACT:
In this series of 3 talks I examine Andrews' spt-function
and how it may be generalized. My main goal is to prove a certain
inequality between crank and rank moments that was only known
previously asymptotically.
TALK 1:
Review of
the rank and crank of partitions, rank and crank moments and
Andrews' spt-function. Andrews porved that the generating function
for the spt-function can be written in terms of the second crank
and rank moment functions. We use limiting form of Bailey's lemma
to derive this result.
TALK 2:
By iterating Bailey's Lemma and using two famous Bailey pairs
we derive generating functions for 2k-th order symmetrized
crank and rank moments as exlicit multiple q-series.
TALK 3:
We use an analog of Stirling numbers of the second kind
to write ordinary moments in terms of symmetrized moments.
We then use our results for symmetrized moments to derive
the following crank-rank-moment inequality
M2k(n) >
N2k(n),
for all n and k greater than 0.
This result had ony been known earlier asymptotically.
We define a weighted partition function and thus give
a combinatorial definition of a higher-spt function sptk(n)
that satisfies
sptk(n)
= μ2k(n) – η2k(n).
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