165 research outputs found
The spt-crank for overpartitions
Bringmann, Lovejoy, and Osburn showed that the generating functions of the
spt-overpartition functions spt(n), spt1(n), spt2(n), and M2spt(n) are
quasimock theta functions, and satisfy a number of simple Ramanujan-like
congruences. Andrews, Garvan, and Liang defined an spt-crank in terms of
weighted vector partitions which combinatorially explain simple congruences mod
5 and 7 for spt (n). Chen, Ji, and Zang were able to define this spt-crank in
terms of ordinary partitions. In this paper we define spt-cranks in terms of
vector partitions that combinatorially explain the known simple congruences for
all the spt-overpartition functions as well as new simple congruences. For all
the overpartition functions except M2spt(n) we are able to define the spt-crank
purely in terms of marked overpartitions. The proofs of the congruences depend
on Bailey's Lemma and the difference formulas for the Dyson rank of an
overpartition and the M2-rank of a partition without repeated odd parts
The spt-Crank for Ordinary Partitions
The spt-function was introduced by Andrews as the weighted counting
of partitions of with respect to the number of occurrences of the smallest
part. Andrews, Garvan and Liang defined the spt-crank of an -partition which
leads to combinatorial interpretations of the congruences of mod 5 and
7. Let denote the net number of -partitions of with spt-crank
. Andrews, Garvan and Liang showed that is nonnegative for all
integers and positive integers , and they asked the question of finding
a combinatorial interpretation of . In this paper, we introduce the
structure of doubly marked partitions and define the spt-crank of a doubly
marked partition. We show that can be interpreted as the number of
doubly marked partitions of with spt-crank . Moreover, we establish a
bijection between marked partitions of and doubly marked partitions of .
A marked partition is defined by Andrews, Dyson and Rhoades as a partition with
exactly one of the smallest parts marked. They consider it a challenge to find
a definition of the spt-crank of a marked partition so that the set of marked
partitions of and can be divided into five and seven equinumerous
classes. The definition of spt-crank for doubly marked partitions and the
bijection between the marked partitions and doubly marked partitions leads to a
solution to the problem of Andrews, Dyson and Rhoades.Comment: 22 pages, 6 figure
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
Partitions with parts separated by parity: conjugation, congruences and the mock theta functions
Noting a curious link between Andrews' even-odd crank and the Stanley rank,
we adopt a combinatorial approach building on the map of conjugation and
continue the study of integer partitions with parts separated by parity. Our
motivation is twofold. First off, we derive results for certain restricted
partitions with even parts below odd parts. These include a Franklin-type
involution proving a parametrized identity that generalizes Andrews' bivariate
generating function, and two families of Andrews--Beck type congruences.
Secondly, we introduce several new subsets of partitions that are stable (i.e.,
invariant under conjugation) and explore their connections with three third
order mock theta functions , , and ,
introduced by Ramanujan and Watson.Comment: 20 pages, 6 figure
Mock theta functions and weakly holomorphic modular forms modulo 2 and 3
We prove that the coefficients of certain mock theta functions possess no
linear congruences modulo 3. We prove similar results for the moduli 2 and 3
for a wide class of weakly holomorphic modular forms and discuss applications.
This extends work of Radu on the behavior of the ordinary partition function
modulo 2 and 3.Comment: 19 page
Mock Jacobi forms in basic hypergeometric series
We show that some -series such as universal mock theta functions are
linear sums of theta quotients and mock Jacobi forms of weight 1/2, which
become holomorphic parts of real analytic modular forms when they are
restricted to torsion points and multiplied by suitable powers of . And we
prove that certain linear sums of -series are weakly holomorphic modular
forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by
mock Jacobi forms. As an application, we obtain a relation between the rank and
crank of a partition.Comment: 13 page
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