George E. Andrews
YOU?
Author Swipe
View article: A refinement of the Crank-Mex theorem
A refinement of the Crank-Mex theorem Open
It is proved that the number of partitions of n with odd mex and k parts that aren’t ones equals the number of partitions of n with nonnegative crank and k parts that aren’t ones.
View article: Certain positive $q$-series and inequalities for two-color partitions
Certain positive $q$-series and inequalities for two-color partitions Open
We consider some $q$-series which depend on a pair of positive integers $(k,m)$. While positivity of these series holds for the first few values of $(k,m)$, the situation is quite unclear for other values of $(k,m)$. In addition, our serie…
View article: Positive Weighted Partitions Generated by Double Series
Positive Weighted Partitions Generated by Double Series Open
We investigate some weighted integer partitions whose generating functions are double-series. We will establish closed formulas for these $q$-double series and deduce that their coefficients are non-negative. This leads to inequalities amo…
View article: On a formula of the $q$-series $_{2k+4}ϕ_{2k+3}$ and its applications
On a formula of the $q$-series $_{2k+4}ϕ_{2k+3}$ and its applications Open
In this paper we apply a formula of the very-well poised $_{2k+4}ϕ_{2k+3}$ to write a $k$-tuple sum of $q$-series as a linear combination of terms wherein each term is a product of expressions of the form $\frac{1}{(qy, qy^{-1};q)_\infty}$…
View article: Legendre theorems for certain overpartitions and overpartition pairs
Legendre theorems for certain overpartitions and overpartition pairs Open
Motivated by two Legendre-type formulas for overpartitions, we derive a variety of their companions as Legendre theorems for overpartition pairs. This leads to equalities of subclasses of overpartitions and overpartition pairs.
View article: On two-color partitions with odd smallest part
On two-color partitions with odd smallest part Open
We will consider two-color integer partitions in which the smallest part is odd and the odd or the even parts may occur only in one color. It turns out that these partitions are generated by the mock theta functions of third order due to R…
View article: Identities involving partitions with distinct even parts and $4$-regular partitions
Identities involving partitions with distinct even parts and $4$-regular partitions Open
It is well known that the number of partitions into distinct even parts equals the number of $4$-regular partitions. In this paper we prove identities relating certain restricted partitions into distinct even parts with restricted $4$-regu…
View article: Further study on MacMahon-type sums of divisors
Further study on MacMahon-type sums of divisors Open
This paper is devoted to the study of $$ U_t(a,q):=\sum_{1\leq n_1
View article: Hook lengths in self-conjugate partitions
Hook lengths in self-conjugate partitions Open
In 2010, G.-N. Han obtained the generating function for the number of size hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even . If is the numbe…
View article: MacMahon's partition analysis XV: Parity
MacMahon's partition analysis XV: Parity Open
We apply the methods of partition analysis to partitions in which the parity of parts plays a role. We begin with an in-depth treatment of the generating function for the partitions from the first Göllnitz-Gordon identity. We then deduce a…
View article: Hook lengths in self-conjugate partitions
Hook lengths in self-conjugate partitions Open
In 2010, G.-N. Han obtained the generating function for the number of size $t$ hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even $t$. If $n_t(λ)…
View article: Extensions of MacMahon's sums of divisors
Extensions of MacMahon's sums of divisors Open
In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so we reveal a wealth of divisibility theorems and u…
View article: Linked partition ideals and a family of quadruple summations
Linked partition ideals and a family of quadruple summations Open
Recently, $4$-regular partitions into distinct parts are connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews for the two types of partitions. Such an extension is made po…
View article: Sequences in overpartitions
Sequences in overpartitions Open
This paper is devoted to the study of sequences in overpartitions and their relation to 2-color partitions. An extensive study of a general class of double series is required to achieve these ends.
View article: On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-measure of partitions and distinct partitions
On the -measure of partitions and distinct partitions Open
The -measure of an integer partition was recently introduced by Andrews, Bhattacharjee and Dastidar. In this paper, we establish trivariate generating function identities counting both the length and the -measure for partitions and distinc…
View article: Separable integer partition classes
Separable integer partition classes Open
A classical method for partition generating function is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with copies of are presented.
View article: ON THE PARITY OF THE GENERALISED FROBENIUS PARTITION FUNCTIONS
ON THE PARITY OF THE GENERALISED FROBENIUS PARTITION FUNCTIONS Open
Andrews [ Generalized Frobenius Partitions , Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions, $\phi _k(n)$ and $c\phi _k(n),$ enumerating two types …
View article: Hook length and symplectic content in partitions
Hook length and symplectic content in partitions Open
The dimension of an irreducible representation of $GL(n,\mathbb{C})$, $Sp(2n)$, or $SO(n)$ is given by the respective hook-length and content formulas for the corresponding partition. The first author, inspired by the Nekrasov-Okounkov for…
View article: Refinements of Beck-type partition identities
Refinements of Beck-type partition identities Open
Franklin's identity generalizes Euler's identity and states that the number of partitions of $n$ with $j$ different parts divisible by $r$ equals the number of partitions of $n$ with $j$ repeated parts. In this article, we give a refinemen…
View article: The Singular Support of the Ising Model
The Singular Support of the Ising Model Open
We prove a new quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson–Slater $q$-series identity of Rogers–Ramanujan type and to Nahm sums for the matrix $\left (\begin {smallmatrix}8&3\\3&…
View article: Schmidt-type theorems for partitions with uncounted parts
Schmidt-type theorems for partitions with uncounted parts Open
Schmidt's theorem is significantly generalized, to partitions in which periodic but otherwise arbitrary subsets of parts are counted or uncounted. The identification of such sets of partitions with colored partitions satisfying certain spe…
View article: Partition Identities for Two-Color Partitions
Partition Identities for Two-Color Partitions Open
Three new partition identities are found for two-color partitions. The first relates to ordinary partitions into parts not divisible by 4, the second to basis partitions, and the third to partitions with distinct parts. The surprise of the…
View article: On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$
On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$ Open
In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $ϕ_k(n)$ and $cϕ_k(n),$ which enumerate two types of combinatorial objects which Andrews called generalized Frobenius partitions. As…
View article: Sequences in Overpartitions
Sequences in Overpartitions Open
This paper is devoted to the study of sequences in overpartitions and their relation to 2-color partitions. An extensive study of a general class of double series is required to achieve these ends.
View article: Linked partition ideals and the Alladi--Schur theorem
Linked partition ideals and the Alladi--Schur theorem Open
Let $\mathscr{S}$ denote the set of integer partitions into parts that differ by at least $3$, with the added constraint that no two consecutive multiples of $3$ occur as parts. We derive trivariate generating functions of Andrews--Gordon …