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George E. Andrews

Publications and source records attributed to George E. Andrews.

At least 19 recordsLinked to original sources
arXiv2026

A surprising generalization of the Möbius function

We introduce a broad generalization of a recursive formula for the Möbius function $μ(n) = 1-n-\sum_{d=2}^{n-1}μ(d)\left[\frac{n}{d}\right]$ due to George Spencer-Brown. By replacing the greatest integer function $\left[\frac{n}{d}\right]$ in this classical recurrence with an arbitrary arithmetic function $f\left(\left[\frac{n}{d}\right]\right)$ with $f(1)=1$, we define a new generalized family of functions, denoted $\star(n)$. We prove that $\star(p) = -1$ if and only if $p$ is prime. This result yields a surprising algebraic characterization of primes and reveals a deep structural property underlying divisor sums and the greatest integer function.

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arXiv2026

On a two-color partition series and its companions

We study the two-color distinct-part series \(S_1(q)\), equivalently Andrews' generating function \(v_d(q)\) for strictly concave compositions, and its odd and even companions \(T_o(q)\) and \(T_e(q)\). We determine the coefficients of \(S_1(q)\) modulo \(4\) and obtain a complete criterion for the resulting Ramanujan-type progressions. For the even companion, we give a direct overpartition interpretation of its coefficients and show that two natural partition families are each counted by half of those coefficients. For the eta-normalized odd companion \(C(q)=(q;q)_\infty T_o(q)\), we prove a quintic self-similarity, derive exact vanishing relations and infinite sign changes for its coefficients, and show that \(c(n)\) can be nonzero only when \(24n+28\) is represented by \(x^2+3y^2\).

math.CO↗
arXiv2026

Weighted partitions with interval restrictions: exact formulas and a bivariate master identity

Let $a_2''(n)$ and $b_2''(n)$ be the signed partition functions introduced by Andrews and El Bachraoui for interval-restricted partitions whose parts greater than $1$ are controlled by the smallest even part and by the number of ones. We prove two conjectures for these functions. The first gives the generating function for $a_2''(n)$ as an elementary rational term plus a false theta series with periodic signs; the second asserts that the companion coefficients $b_2''(n)$ take only the values $-1,0,1,2$. The central structural result introduces an auxiliary variable $z$ recording the number of non-compulsory parts greater than $1$. We obtain closed forms for the two resulting generating functions and prove the master identity $(1+q^2)\mathcal B(z,q)-(1+q)\mathcal A(z,q)=-q^4/(1-q^3)$ using both analytic and combinatorial techniques. At $z=-1$, this identity, together with a Rogers--Fine evaluation, gives the false theta formula for $a_2''(n)$ and an explicit generating function for $b_2''(n)$. The latter formula implies the asserted coefficient range and leads to an exact coefficient description of $b_2''(n)$. We also include a direct Heine--Rogers--Fine proof of the false theta formula, ordinary and fixed-refinement consequences of the master identity, and the resulting quantum modular interpretation.

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arXiv2026

Two-color partitions with evens in one color

We consider sequences counting integer partitions in two colors (red and blue) in which the even parts occur only in blue color. We focus on subsequences defined by constraints on the parity and color of the summands. We establish formulas for our sequences and deduce identities of integer partitions.

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arXiv2026

Solutions for Hecke Sum Questions of Banerjee and Bringmann

The present authors introduced a two-color partition series $S(q)$ and conjectured a Hecke-type formula for the even part of $(q^4;q^4)_\infty S(q)$. Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular completions, and Sturm's theorem. They also asked whether a direct proof, for instance one based on Bailey-type ideas, could be found, and they suggested that the odd residue classes may be worth studying. We prove a two-variable refinement with an additional parameter $a$. Our proof relies entirely on $q$-series combined with the Bailey pairs The original even identity and the odd identity then follow as corollaries by letting $a=1$. We also record parameter symmetries and cyclotomic companions, including a vanishing result at $a=i$.

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arXiv2026

On Glaisher's Partition Theorem

Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m=2$ case is Euler's famous partition theorem. Recently, Andrews, Kumar, and Yee gave two new partition functions $C(n)$ and $D(n)$ related to Euler's theorem. Lin and Zang extended their result to Glaisher's theorem by generalizing $C(n)$. We generalize $D(n)$ and prove an analogous partition identity for the $m=3$ case. We also provide a new series equal to Glaisher's product both in the finite and infinite cases.

math.CO↗
arXiv2026

Congruences for two-color partitions with odd smallest part

For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)\pmod{4}$ and obtain congruences modulo $2$ and $4$ for $C(k,n)$ when $k=2,3$. Using $q$-series methods we derive closed formulas for $C_k(q)$ in terms of eta-quotients and formulate Ramanujan-type congruences for the limiting sequence arising from $\lim_{k\to\infty} C_k(q)$.

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arXiv2025

Extending Andrews and Newman's refinement of the crank-mex theorem

The crank-mex theorem states that the number of integer partitions of $n$ with nonnegative crank equals the number with odd minimal excludant (mex). Andrews and M. Newman recently refined that result in terms of the number of parts greater than one. Here, we establish and expand a complementary result connecting the partitions with even mex, having fixed points, with negative crank, and with positive crank, all refined in terms of number of parts greater than one. We provide both analytic and combinatorial proofs.

math.CO↗
arXiv2025

On Euler's Theorem

Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*} A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1), \end{align*} where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.

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arXiv2025

Certain positive $q$-series and inequalities for two-color partitions

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 series generate the number of certain two-color integer partitions weighted by $(-1)^j$ where $j$ is the number of even parts. Therefore, inequalities involving these partitions will be deduced from the positivity of their generating functions.

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arXiv2025

Positive Weighted Partitions Generated by Double Series

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 among integer partitions.

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arXiv2025

Lambert series and double Lambert series

We consider relationships between classical Lambert series, multiple Lambert series and classical $q$-series of the Rogers-Ramanujan type. We conclude with a contemplation on the Andrews-Dixit-Schultz-Yee conjecture.

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arXiv2025

On a formula of the $q$-series $_{2k+4}ϕ_{2k+3}$ and its applications

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 $\frac{1}{(qy, qy^{-1};q)_\infty}$. As an application, we shall express a variety of sums and double sums of $q$-series as linear combinations of infinite products. Our formulas are motivated by their connection to overpartition pairs.

math.CO↗
arXiv2025

Rank, two-color partitions and Mock theta function

In this paper, we establish that the number of partitions of a natural number with positive odd rank is equal to the number of two-color partitions (red and blue), where the smallest part is even (say $2n$) and all red parts are even and lie within the interval $(2n,4n]$. This led us to derive a new representation for the third order mock theta function $f_3(q)$ and an analogue of the fundamental identity for the smallest part partition function Spt$(n)$, both of which are of significant interest in their own right. We also consider the odd smallest part version of the above two-color partition, whose generating function involves another third order mock theta function $ϕ_3(q)$.

math.CO↗
arXiv2024

Further study on MacMahon-type sums of divisors

This paper is devoted to the study of $$ U_t(a,q):=\sum_{1\leq n_1<n_2<\cdots<n_t}\frac{q^{n_1+n_2+\cdots+n_t}}{(1+aq^{n_1}+q^{2n_1})(1+aq^{n_2}+q^{2n_2})\cdots(1+aq^{n_t}+q^{2n_t})} $$ when $a$ is one of $0, \pm 1, \pm2$. The idea builds on our previous treatment of the case $a=-2$. It is shown that all these functions lie in the ring of quasi-modular forms. Among the more surprising findings is $$U_2(1,q)=\sum_{n\geq1} \frac{q^{3n}}{(1-q^{3n})^2}.$$

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arXiv2024

Hook lengths in self-conjugate partitions

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 $n_t(λ)$ is the number of size $t$ hooks in a partition $λ,$ then for even $t$ we have $$\sum_{λ\in \mathcal{SC}} x^{n_t(λ)} q^{\vertλ\vert} = (-q;q^2)_{\infty} \cdot ((1-x^2)q^{2t};q^{2t})_{\infty}^{\frac{t}2}. $$ As a consequence, if $a_t^*(n)$ is the number of such hooks among the self-conjugate partitions of $n,$ then for even $t$ we obtain the simple formula $$ a_t^*(n)=t\sum_{j\geq 1} q^*(n-2tj), $$ where $q^*(m)$ is the number of partitions of $m$ into distinct odd parts. As a corollary, we find that $t\mid a_t^*(n),$ which confirms a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.

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