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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…

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..

Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer that is not a part. The concept has been used since at least 2011, though, with connections to Frobenius symbols. Recently the…

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 statistics, the mex, and…

Let $k$ be a positive integer and $m$ be an integer. Garvan's $k$-rank $N_k(m,n)$ is the number of partitions of $n$ into at least $(k-1)$ successive Durfee squares with $k$-rank equal to $m$. In this paper give some asymptotics for…

Euler's classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. Franklin gave a generalization by considering partitions with exactly $j$ different multiples of $r$, for a…

The notion of the spt-crank of a vector partition, or an $S$-partition, was introduced by Andrews, Garvan and Liang. Let $N_S(m,n)$ denote the number of $S$-partitions of $n$ with spt-crank $m$. Andrews, Dyson and Rhoades conjectured that…

Euler's classic partition identity states that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We develop a new generalization of this identity, which yields a previous…

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…

Let $M_0(n)$ (resp. $M_1(n)$) denote the number of partitions of $n$ with even (reps. odd) crank. Choi, Kang and Lovejoy established an asymptotic formula for $M_0(n)-M_1(n)$. By utilizing this formula with the explicit bound, we show that…

In this paper we refine a weighted partition identity of Alladi. We write explicit formulas of generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results…

Denote by $p(n)$ the number of partitions of $n$ and by $N(a,M;n)$ the number of partitions of $n$ with rank congruent to $a$ modulo $M$. By considering the deviation \begin{equation*} D(a,M) := \sum_{n= 0}^{\infty}\left(N(a,M;n) -…

In this paper, we discuss a few recent conjectures made by George Beck related to the ranks and cranks of partitions. The conjectures for the rank of a partition were proved by Andrews by using results due to Atkin and Swinnerton-Dyer on a…

In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck.…

In the preceding decade, Andrews and Newman resurrected the concept of a `minimal excludant' of a partition ($mex$, for short), namely, the least positive missing integer in a partition. Subsequently, several authors have not only studied…

In 2021, Andrews mentioned that George Beck introduced partition statistics $M_w(r,m,n)$, which denote the total number of ones in the partition of $n$ with crank congruent to $r$ modulo $m$. Recently, a number of congruences and identities…

Andrews and Merca [J. Combin. Theory Ser. A 203 (2024), Art. 105849] recently obtained two interesting results on the sum of the parts with the same parity in the partitions of $n$ (the modulo $2$ case), the proof of which relies on…

A partition statistic ` crank' gives combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula, Ramanujan type congruences, and q-series identities that the number of…

In this paper, we show that the difference between the number of parts in the odd partitions of $n$ and the number of parts in the distinct partitions of $n$ satisfies Euler's recurrence relation for the partition function $p(n)$ when $n$…

In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer $n$ with even crank and those with odd crank, denoted $C(n)=c_e(n)-c_o(n)$. Inspired by Ramanujan's classical…