DATE:
Tuesday, September 27 (2011), at 1:55pm
PLACE: LIT 368
SPEAKER:
Frank Garvan
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TITLE:
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Combinatorial Interpretations of Congruences for the
SPT-Function - Part 3
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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)
Two weeks ago, 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.
Last week in Part 2, we proved that all the spt-crank coefficients
NS(m,n) are nonnegative. The proof uses several identities
from the theory of basic hypergeometric series.
This week in Part 3, we prove that the number of self-conjugate vector
partitions in S counted with the weight ω1 occur
as the coefficients of a mock theta function studied earlier
by Andrews, Dyson and Hickerson. These coefficients can expressed
as the number of solutions of a binary indefinite quadratic
form (counted in a certain sense) associated with Q(sqrt(6)). This
leads to an elementary q-series proof of a result for the parity
of spt(n), which was found earlier by Folsom and Ono but stated incorrectly.
This is joint work with George Andrews and Jie Liang.
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