43 research outputs found
An overpartition analogue of the -binomial coefficients
We define an overpartition analogue of Gaussian polynomials (also known as
-binomial coefficients) as a generating function for the number of
overpartitions fitting inside the rectangle. We call these new
polynomials over Gaussian polynomials or over -binomial coefficients. We
investigate basic properties and applications of over -binomial
coefficients. In particular, via the recurrences and combinatorial
interpretations of over q-binomial coefficients, we prove a Rogers-Ramaujan
type partition theorem.Comment: v2: new section added about another way of proving our theorems using
q-series identitie
Theorems, Problems and Conjectures
These notes are designed to offer some (perhaps new) codicils to related
work, a list of problems and conjectures seeking (preferably) combinatorial
proofs. The main items are Eulerian polynomials and hook/contents of Young
diagram, mostly on the latter. The new additions include items on Frobenius
theorem and multi-core partitions; most recently, some problems on (what we
call) colored overpartitions. Formulas analogues to or in the spirit of works
by Han, Nekrasov-Okounkov and Stanley are distributed throughout. Concluding
remarks are provided at the end in hopes of directing the interested
researcher, properly.Comment: 14 page
Rademacher-Type Formulas for Partitions and Overpartitions
A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of Ο4_0 | Ο_β1 is presented
Rademacher-Type Formulas for Restricted Partition and Overpartition Functions
A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes
Some Separable integer partition classes
Recently, Andrews introduced separable integer partition classes and analyzed
some well-known theorems. In this paper, we investigate partitions with parts
separated by parity introduced by Andrews with the aid of separable integer
partition classes with modulus . We also extend separable integer partition
classes with modulus to overpartitions, called separable overpartition
classes. We study overpartitions and the overpartition analogue of
Rogers-Ramanujan identities, which are separable overpartition classes