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View article: q-Appell functions and Kleshchev 2-multipartitions
q-Appell functions and Kleshchev 2-multipartitions Open
Two double sums originating from the Ariki–Koike algebras are evaluated and generalized using a transformation of F. H. Jackson. Two combinatorial approaches to these identities are given, using 2-cores and overpartitions.
View article: The Littlewood decomposition via colored Frobenius partitions
The Littlewood decomposition via colored Frobenius partitions Open
The Littlewood decomposition for partitions is a well-known bijection between partitions and pairs of $t$-core and $t$-quotient partitions. This decomposition can be described in several ways, such as the $t$-abacus method of James or the …
View article: The Ariki--Koike algebras and Rogers--Ramanujan type partitions
The Ariki--Koike algebras and Rogers--Ramanujan type partitions Open
In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike algebras $\mathcal{H}_{\mathbb{C},q;Q_1,\ldots, Q_m}\big(G(m, 1, n)\big)$ (when the parameters are roots of unity and $q\neq 1$) are labeled by the so-called Kles…
View article: Combinatorial Perspectives on the Crank and Mex Partition Statistics
Combinatorial Perspectives on the Crank and Mex Partition Statistics Open
Several authors have recently considered the smallest positive part missing from an integer partition, known as the minimum excludant or mex. In this work, we revisit and extend connections between Dyson's crank statistic, the mex, and Fro…
View article: Diagonal Hooks and a Schmidt-Type Partition Identity
Diagonal Hooks and a Schmidt-Type Partition Identity Open
In a recent paper of Andrews and Paule, several Schmidt-type partition identities are considered within the framework of MacMahon's Partition Analysis. Following their work, we derive a new Schmidt-type identity concerning diagonal hooks o…
View article: Schmidt Type Partitions
Schmidt Type Partitions Open
Recently, Andrews and Paule studied Schmidt type partitions using MacMahon's Partition Analysis and obtained various interesting results. In this paper, we focus on the combinatorics of Schmidt type partition theorems and characterize them…
View article: Schmidt type partitions
Schmidt type partitions Open
Recently, Andrews and Paule studied Schmidt-type partitions using MacMahon's Partition Analysis and obtained various interesting results.In this paper, we focus on the combinatorics of Schmidt-type partition theorems and characterize them …
View article: Generating Functions for Certain Weighted Cranks
Generating Functions for Certain Weighted Cranks Open
Recently, George Beck posed many interesting partition problems considering the number of ones in partitions. In this paper, we first consider the crank generating function weighted by the number of ones and obtain analytic formulas for th…
View article: On the sum of parts with multiplicity at least 2 in all the partitions of n
On the sum of parts with multiplicity at least 2 in all the partitions of n Open
In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of [Formula: see text] providing new combinatorial interpretations for this sum. A connection with subsets of [Formula: see text] is…
View article: Index of Seaweed Algebras and Integer Partitions
Index of Seaweed Algebras and Integer Partitions Open
The index of a Lie algebra is an important algebraic invariant. In 2000, Vladimir Dergachev and Alexandre Kirillov defined seaweed subalgebras of $\mathfrak{gl}_n$ (or $\mathfrak{sl}_n$) and provided a formula for the index of a seaweed al…
View article: Index of seaweed algebras and integer partitions
Index of seaweed algebras and integer partitions Open
The index of a Lie algebra is an important algebraic invariant. In 2000, Vladimir Dergachev and Alexandre Kirillov defined seaweed subalgebras of $\mathfrak{gl}_n$ (or $\mathfrak{sl}_n$) and provided a formula for the index of a seaweed al…
View article: A lecture hall theorem for $m$-falling partitions
A lecture hall theorem for $m$-falling partitions Open
For an integer $m\ge 2$, a partition $λ=(λ_1,λ_2,\ldots)$ is called $m$-falling, a notion introduced by Keith, if the least nonnegative residues mod $m$ of $λ_i$'s form a nonincreasing sequence. We extend a bijection originally due to the …
View article: Some Identities associated with mock theta functions $ω(q)$ and $ν(q)$
Some Identities associated with mock theta functions $ω(q)$ and $ν(q)$ Open
Recently, Andrews, Dixit and Yee defined two partition functions $p_ω(n)$ and $p_ν(n)$ that are related with Ramanujan's mock theta functions $ω(q)$ and $ν(q)$, respectively. In this paper, we present two variable generalizations of their …
View article: Some Identities associated with mock theta functions $\omega(q)$ and $\nu(q)$
Some Identities associated with mock theta functions $\omega(q)$ and $\nu(q)$ Open
Recently, Andrews, Dixit and Yee defined two partition functions $p_{\omega}(n)$ and $p_{\nu}(n)$ that are related with Ramanujan's mock theta functions $\omega(q)$ and $\nu(q)$, respectively. In this paper, we present two variable general…
View article: Enumeration of artitions with prescribed successive rank parity blocks
Enumeration of artitions with prescribed successive rank parity blocks Open
Successive ranks of a partition, which were introduced by Atkin, are the difference of the $i$th row and the $i$th column in the Ferrers graph. Recently, in the study of singular overpartitions, Andrews revisited successive ranks and parit…
View article: Overpartitions and singular overpartitions
Overpartitions and singular overpartitions Open
Singular overpartitions, which are defined by George Andrews, are overpartitions whose Frobenius symbols have at most one overlined entry in each row. In his paper, Andrews obtained interesting combinatorial results on singular overpartiti…
View article: Overpartitions related to the mock theta function $\omega (q)$
Overpartitions related to the mock theta function $\omega (q)$ Open
It was recently shown that $q\\omega(q)$, where $\\omega(q)$ is one of the\nthird order mock theta functions, is the generating function of\n$p_{\\omega}(n)$, the number of partitions of a positive integer $n$ such that\nall odd parts are …
View article: Overpartitions related to the mock theta function $ω(q)$
Overpartitions related to the mock theta function $ω(q)$ Open
It was recently shown that $qω(q)$, where $ω(q)$ is one of the third order mock theta functions, is the generating function of $p_ω(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less than twice the sma…
View article: Partitions associated with the Ramanujan/Watson mock theta functions ω(q), ν(q)and ϕ(q)
Partitions associated with the Ramanujan/Watson mock theta functions ω(q), ν(q)and ϕ(q) Open
The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function ω(q) (resp. ν(−q)). Similar results for partitions w…
View article: Partitions associated with the Ramanujan/Watson mock theta functions $ω(q), ν(q)$ and $ϕ(q)$
Partitions associated with the Ramanujan/Watson mock theta functions $ω(q), ν(q)$ and $ϕ(q)$ Open
The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function $ω(q)$ (resp. $ν(-q)$). Similar results for partitio…
View article: Partitions associated with the Ramanujan/Watson mock theta functions $\omega(q), \nu(q)$ and $\phi(q)$
Partitions associated with the Ramanujan/Watson mock theta functions $\omega(q), \nu(q)$ and $\phi(q)$ Open
The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function $\omega(q)$ (resp. $\nu(-q)$). Similar results for p…