6,298 research outputs found
Peak positions of strongly unimodal sequences
We study combinatorial and asymptotic properties of the rank of strongly
unimodal sequences. We find a generating function for the rank enumeration
function, and give a new combinatorial interpretation of the ospt-function
introduced by Andrews, Chan, and Kim. We conjecture that the enumeration
function for the number of unimodal sequences of a fixed size and varying rank
is log-concave, and prove an asymptotic result in support of this conjecture.
Finally, we determine the asymptotic behavior of the rank for strongly unimodal
sequences, and prove that its values (when appropriately renormalized) are
normally distributed with mean zero in the asymptotic limit
Asymptotics for rank and crank moments
Moments of the partition rank and crank statistics have been studied for
their connections to combinatorial objects such as Durfee symbols, as well as
for their connections to harmonic Maass forms. This paper proves a conjecture
due to Bringmann and Mahlburg that refined a conjecture of Garvan. Garvan's
conjecture states that the moments of the crank function are always larger than
the moments of the rank function, even though the moments have the same main
asymptotic term. The proof uses the Hardy-Ramanujan method to provide precise
asymptotic estimates for rank and crank moments and their differences.Comment: 11 page
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