228 research outputs found
Ramanujan type identities and congruences for partition pairs
AbstractUsing elementary methods, we establish several new Ramanujan type identities and congruences for certain pairs of partition functions
Arithmetic Properties of Overpartition Pairs
Bringmann and Lovejoy introduced a rank for overpartition pairs and
investigated its role in congruence properties of , the number of
overpartition pairs of n. In particular, they applied the theory of Klein forms
to show that there exist many Ramanujan-type congruences for the number
. In this paper, we shall derive two Ramanujan-type identities and
some explicit congruences for . Moreover, we find three ranks as
combinatorial interpretations of the fact that is divisible by
three for any n. We also construct infinite families of congruences for
modulo 3, 5, and 9.Comment: 19 page
Some Smallest Parts Functions from Variations of Bailey's Lemma
We construct new smallest parts partition functions and smallest parts crank
functions by considering variations of Bailey's Lemma and conjugate Bailey
pairs. The functions we introduce satisfy simple linear congruences modulo
and . We introduce and give identities for two four variable
-hypergeometric functions; these functions specialize to some of our new
spt-crank-type functions as well as many known spt-crank-type functions
Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall-Littlewood Polynomials
Here we consider the -series coming from the Hall-Littlewood polynomials,
\begin{equation*} R_\nu(a,b;q)=\sum_{\substack{\lambda \\[1pt] \lambda_1\leq
a}} q^{c|\lambda|} P_{2\lambda}\big(1,q,q^2,\dots;q^{2b+d}\big).
\end{equation*} These series were defined by Griffin, Ono, and Warnaar in their
work on the framework of the Rogers-Ramanujan identities. We devise a recursive
method for computing the coefficients of these series when they arise within
the Rogers-Ramanujan framework. Furthermore, we study the congruence properties
of certain quotients and products of these series, generalizing the famous
Ramanujan congruence \begin{equation*} p(5n+4)\equiv0\pmod{5}. \end{equation*}Comment: 16 pages v2: Minor changes included, to appear in Annals of
Combinatoric
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