Narrow Results

Let $G$ be a finite group. For a given pair of non-negative integers $n$ and $k$, we aim to give an explicit formula to count the number of elements of ${\mathrm{\Lambda }}_k(G)$, the set of all irreducible representations of $G$ whose degree is not divisible by $2^k$. In this paper, we give formulas for $|{\mathrm{\Lambda }}_k(G)|$, when $G$ is symmetric group, generalized symmetric group, and alternating group. Further, we also have a complete characterization of the elements in ${\mathrm{\Lambda }}_2(S_n)$, ${\mathrm{\Lambda }}_1(G(n,r))$, and the self-conjugate partitions of $2^l$ and $2^l+1$ with degree not divisible by $4$.

It is shown that (two-variable generalizations of) more than half of Slater's list of 130 Rogers–Ramanujan identities (L. J. Slater, Further identities of the Rogers–Ramanujan type, Proc. London Math Soc. (2)54 (1952) 147–167) can be easily derived using just three multiparameter Bailey pairs and their associated q-difference equations. As a bonus, new Rogers–Ramanujan type identities are found along with natural combinatorial interpretations for many of these identities.

In this paper, we prove a theorem of Fine bijectively. Stacks with summits and gradual stacks with summits are also discussed.

We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

We describe an extension of Fine’s method of deriving basic hypergeometric series transformations and derive new transformations from the method. Combinatorial proofs of two of the examples are provided.

In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of $n$ providing new combinatorial interpretations for this sum. A connection with subsets of $\{1,2,\dots ,n\}$ is given in this context. We provide two different proofs of our results: analytic and combinatorial. In addition, considering the multiplicity of parts in a partition, we provide a combinatorial proof of the truncated pentagonal number theorem of Andrews and Merca.

Recently, Merca and Yee proved some partition identities involving two new partition statistics. We refine these statistics and generalize the results of Merca and Yee. We also correct a small mistake in a result of Merca and Yee.

We prove two partition identities which are dual to the Rogers–Ramanujan identities. These identities are inspired by (and proved using) a correspondence between three kinds of objects: a new type of partitions (neighborly partitions), monomial ideals and some infinite graphs.

Recently, Andrews, Bhattacharjee and Dastidar introduced the concept of $k$-measure of an integer partition, and proved a surprising identity that the number of partitions of $n$ which have $2$-measure $m$ is equal to the number of partitions of $n$ with a Durfee square of side $m$. The authors asked for a bijective proof of this result and also suggested a further exploration of the properties of the number of partitions of $n$ which have $k$-measure $m$ for $k\ge 3$. In this paper, we perform these tasks. That is, we obtain a short combinatorial proof of the result of Andrews, Bhattacharjee and Dastidar, and using this proof, we obtain a natural generalization for $k$-measures.

We give new proofs of the 12 Rogers–Ramanujan-type identities due to Rogers and Slater that are traditionally associated with the moduli 7, 14 and 28.

In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas–Eisenstein Series. First, we show that the Pell-Lucas–Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas numbers.

The concept of a modular difference set was originally motivated by the cognate notion of modular Hadamard matrices, which have been researched extensively. We initiate the study of the repetition-parameter set in a modular difference set, and we relate the repetition-parameter set to integer partitions and Diophantine equations. By example, we show how a computational study of integer partitions can improve the upper bound on the size of such repetition-parameter set. All previously known examples of modular difference sets in a direct product of groups are concerned with a product of just two groups. We present new constructions of modular difference sets in a direct product of n groups. These new constructions suggest that the size of the repetition-parameter set is intimately related to the group’s structure. A generalization of difference sets, partial difference sets, and relative difference sets, modular difference sets have been used to construct both modular symmetric designs and equiangular tight frames in finite fields.

We describe an extension of Fine’s method of deriving basic hypergeometric series transformations and derive new transformations from the method. Combinatorial proofs of two of the examples are provided.

In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of n providing new combinatorial interpretations for this sum. A connection with subsets of {1, 2, …, n} is given in this context. We provide two different proofs of our results: analytic and combinatorial. In addition, considering the multiplicity of parts in a partition, we provide a combinatorial proof of the truncated pentagonal number theorem of Andrews and Merca.

We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

For prime levels 5 ≤ p ≤ 19, sets of theta quotients are constructed that generate graded rings of modular forms of positive integer weight for Γ₁(p). Action by Γ₀(p) is shown to cyclically permute the generators. This induces symmetric representations for modular forms. The generators are used to deduce representations for the number of t-core partitions of an integer as convolutions of L-functions. Coupled systems of differential equations of level p are constructed for each basis analogous to Ramanujan’s differential equations for Eisenstein series on the full modular group.