Ranks of partitions play an important role in the theory of partitions. They provide combinatorial interpretations for Ramanujan’s famous congruences for partition functions. We establish a family of congruences modulo powers of $5$ for ranks of partitions.

Noting a curious link between Andrews’ even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. Firstly, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews’ bivariate generating function, and two families of Andrews–Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e. invariant under conjugation) and explore their connections with three third-order mock theta functions $\omega (q)$, $\nu (q)$, and $\psi ^{(3)}(q)$, introduced by Ramanujan and Watson.

In 2019, Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb. 23(2) (2019), 249–254] introduced the arithmetic function $\sigma \textrm {mex}(n)$, which denotes the sum of minimal excludants over all the partitions of n. Baruah et al. [‘A refinement of a result of Andrews and Newman on the sum of minimal excludants’, Ramanujan J., to appear] showed that the sum of minimal excludants over all the partitions of n is the same as the number of partition pairs of n into distinct parts. They proved three congruences modulo $4$ and $8$ for two functions appearing in this refinement and conjectured two further congruences modulo $8$ and $16$. We confirm these two conjectures by using q-series manipulations and modular forms.

For which infinite cardinals $\kappa $ is there a partition of the real line ${\mathbb R}$ into precisely $\kappa $ Borel sets? Work of Lusin, Souslin, and Hausdorff shows that ${\mathbb R}$ can be partitioned into $\aleph _1$ Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of ${\mathbb R}$ into Borel sets can be fairly arbitrary. For example, given any $A \subseteq \omega $ with $0,1 \in A$, there is a forcing extension in which ${A = \{ n :\, \text {there is a partition of } {{\mathbb R}} \text { into }\aleph _n\text { Borel sets}\}}$. We also look at the corresponding question for partitions of ${\mathbb R}$ into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable $\kappa $ such that there is a partition of ${\mathbb R}$ into precisely $\kappa $ closed sets can be fairly arbitrary.

Let $Q(n)$ denote the number of partitions of n into distinct parts. Merca [‘Ramanujan-type congruences modulo 4 for partitions into distinct parts’, An. Şt. Univ. Ovidius Constanţa 30(3) (2022), 185–199] derived some congruences modulo $4$ and $8$ for $Q(n)$ and posed a conjecture on congruences modulo powers of $2$ enjoyed by $Q(n)$. We present an approach which can be used to prove a family of internal congruence relations modulo powers of $2$ concerning $Q(n)$. As an immediate consequence, we not only prove Merca’s conjecture, but also derive many internal congruences modulo powers of $2$ satisfied by $Q(n)$. Moreover, we establish an infinite family of congruence relations modulo $4$ for $Q(n)$.

Recently, when studying intricate connections between Ramanujan’s theta functions and a class of partition functions, Banerjee and Dastidar [‘Ramanujan’s theta functions and parity of parts and cranks of partitions’, Ann. Comb., to appear] studied some arithmetic properties for $c_o(n)$, the number of partitions of n with odd crank. They conjectured a congruence modulo $4$ satisfied by $c_o(n)$. We confirm the conjecture and evaluate $c_o(4n)$ modulo $8$ by dissecting some q-series into even powers. Moreover, we give a conjecture on the density of divisibility of odd cranks modulo 4, 8 and 16.

A partition of a positive integer n is called $\ell $-regular if none of its parts is divisible by $\ell $. Denote by $b_{\ell }(n)$ the number of $\ell $-regular partitions of n. We give a complete characterisation of the arithmetic of $b_{23}(n)$ modulo $11$ for all n not divisible by $11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of $b_{23}(n)$ and certain modular forms having complex multiplication by ${\mathbb Q}(\sqrt {-69})$.

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Motivated by the works of Andrews and Merca, Guo and Zeng deduced truncated versions for two other classical theta series identities of Gauss. Very recently, Xia et al. proved new truncated theorems of the three classical theta series identities by taking different truncated series than the ones chosen by Andrews–Merca and Guo–Zeng. In this paper, we provide a unified treatment to establish new truncated versions for the three identities of Euler and Gauss based on a Bailey pair due to Lovejoy. These new truncated identities imply the results proved by Andrews–Merca, Wang–Yee, and Xia–Yee–Zhao.

Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations that arise from extensions of the Nekrasov–Okounkov hook product formula and from Betti numbers of various Hilbert schemes of n points on ${\mathbb {C}}^2$ . For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty $ . For t-hooks, we prove distributions that are often not equidistributed. The cases where $t\in \{2, 3\}$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products

$$ \begin{align*}F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\mathrm{and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). \end{align*} $$

Let $\mathcal {C}_n =\left [\chi _{\lambda }(\mu )\right ]_{\lambda , \mu }$ be the character table for $S_n,$ where the indices $\lambda $ and $\mu $ run over the $p(n)$ many integer partitions of $n.$ In this note, we study $Z_{\ell }(n),$ the number of zero entries $\chi _{\lambda }(\mu )$ in $\mathcal {C}_n,$ where $\lambda $ is an $\ell $ -core partition of $n.$ For every prime $\ell \geq 5,$ we prove an asymptotic formula of the form

$$ \begin{align*}Z_{\ell}(n)\sim \alpha_{\ell}\cdot \sigma_{\ell}(n+\delta_{\ell})p(n)\gg_{\ell} n^{\frac{\ell-5}{2}}e^{\pi\sqrt{2n/3}}, \end{align*} $$

$\sigma _{\ell }(n)$

$\delta _{\ell }:=(\ell ^2-1)/24,$

$1/\alpha _{\ell }>0$

$L\left (\left ( \frac {\cdot }{\ell } \right ),\frac {\ell -1}{2}\right ).$

$\ell $

$n>\ell ^6/24,$

$\chi _{\lambda }(\mu )=0$

$\lambda $

$\mu $

$\ell $

$Z^*_{\ell }(n)$

$\ell $

$\ell \geq 5$

$$ \begin{align*}Z^*_{\ell}(n)\sim \alpha_{\ell}^2 \cdot \sigma_{\ell}( n+\delta_{\ell})^2 \gg_{\ell} n^{\ell-3}. \end{align*} $$

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 of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ Our goal is to identify an infinite family of values of k such that $\phi _k(n)$ is even for all n in a specific arithmetic progression; in particular, we prove that, for all positive integers $\ell ,$ all primes $p\geq 5$ and all values $r, 0 < r < p,$ such that $24r+1$ is a quadratic nonresidue modulo $p,$

$$ \begin{align*} \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} \end{align*} $$

for all $n\geq 0.$ Our proof of this result is truly elementary, relying on a lemma from Andrews’ memoir, classical q-series results and elementary generating function manipulations. Such a result, which holds for infinitely many values of $k,$ is rare in the study of arithmetic properties satisfied by generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.

Hausel and Rodriguez-Villegas (2015, Astérisque 370, 113–156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $(\mathbb {C}^{2})^{[n]}$ on $n$ points, as $n\rightarrow +\infty ,$ is a Gumbel distribution. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $((\mathbb {C}^{2})^{[n]})^{T_{\alpha ,\beta }}$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $A\geq 2.$ Furthermore, if $p_{k}(A;n)$ denotes the number of partitions of $n$ with exactly $k$ parts that are multiples of $A$ , then we obtain the asymptotic

$$ \begin{align*} p_{k}(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^{k}}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, \end{align*} $$

We investigate the sum of the parts in all the partitions of n into distinct parts and give two infinite families of linear inequalities involving this sum. The results can be seen as new connections between partitions and divisors.

Let r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0,

\[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \]

\[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \]

Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1/(n+1)-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1/2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparison result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

For positive integers $n$ and $k$, let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$, by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$. Furthermore, in view of these arithmetic properties of $r_{k}(n)$, we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

In this paper, we investigate $\unicode[STIX]{x1D70B}(m,n)$, the number of partitions of the bipartite number$(m,n)$ into steadily decreasing parts, introduced by Carlitz [‘A problem in partitions’, Duke Math. J.30 (1963), 203–213]. We give a relation between $\unicode[STIX]{x1D70B}(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Using this relation, we establish some uniform asymptotic formulas for $\unicode[STIX]{x1D70B}(m,n)$.

Let $\unicode[STIX]{x1D707}(m,n)$ (respectively, $\unicode[STIX]{x1D702}(m,n)$) denote the number of odd-balanced unimodal sequences of size $2n$ and rank $m$ with even parts congruent to $2\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ (respectively, $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$) and odd parts at most half the peak. We prove that two-variable generating functions for $\unicode[STIX]{x1D707}(m,n)$ and $\unicode[STIX]{x1D702}(m,n)$ are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single $C^{\infty }$ function in $\mathbb{R}\times \mathbb{R}$ to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables $w$ and $q$, how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size $2n$ with even parts congruent to $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ and odd parts at most half the peak.