DATE:
Tuesday, September 20 (2011), at 1:55pm
PLACE: LIT 368
SPEAKER:
Frank Garvan
|
TITLE:
|
Combinatorial Interpretations of Congruences for the
SPT-Function - Part 2
|
ABSTRACT:
Let spt(n) denote to total number of smallest parts in the
partitions of n. Andrews (2008) proved that
spt(5n + 4) = 0 (mod 5)
spt(7n + 5) = 0 (mod 7)
spt(13n + 6) = 0 (mod 13)
Last week, in Part 1,
we gave a vector-partition type crank which refines the congruences
mod 5 and 7. Let NS(m,n) denote the number of
vector partitions of n in the set S counted with the weight ω1.
In Part 2, we prove that all the spt-crank coefficients
NS(m,n) are nonnegative. The proof uses several identities
from the theory of basic hypergeometric series.
This is joint work with George Andrews and Jie Liang.
|