相关论文: Dyson's Crank and the Mex of Integer 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…

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let $\sigma_o\text{mex}(n)$ (resp., $\sigma_e\text{mex}(n)$) denote the sum of odd (resp., even) minimal excludants over all the…

The Dyson rank of an integer partition is the difference between its largest part and the number of parts it contains. Using Fine-Dyson symmetry, we give formulas for the number of partitions of n with rank larger than n/2, and we prove…

The spt-function $spt(n)$ was introduced by Andrews as the weighted counting of partitions of $n$ with respect to the number of occurrences of the smallest part. Andrews, Garvan and Liang defined the spt-crank of an $S$-partition which…

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…

For each nonempty integer partition $\pi$, we define the maximal excludant of $\pi$ to be the largest nonnegative integer smaller than the largest part of $\pi$ that is not a part of $\pi$. Let $\sigma\!\operatorname{maex}(n)$ be the sum of…

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…

Dyson famously provided combinatorial explanations for Ramanujan's partition congruences modulo $5$ and $7$ via his rank function, and postulated that an invariant explaining all of Ramanujan's congruences modulo $5$, $7$, and $11$ should…

Using quasimodular forms with respect to $\Gamma_0(4)$ we find exact relations between the M2-rank for partitions without repeated odd parts and three residual cranks. From these identities we are able to deduce various congruences mod 3…

Inspired by the study of the minimal excludant in integer partitions by G.E. Andrews and D. Newman, we introduce a pair of new partition statistics, sqrank and rerank. They are related to a polynomial bosonic form of statistical…

In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of 5n+4 into five equal classes. This gave a combinatorial explanation of…

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank <= k and those with k in…

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…

The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite…

Let $N(\leq m,n)$ denote the number of partitions of $n$ with rank not greater than $m$, and let $M(\leq m,n)$ denote the number of partitions of $n$ with crank not greater than $m$. Bringmann and Mahlburg observed that $N(\leq m,n)\leq…

In 2003, Hammond and Lewis defined a statistic on partitions into 2 colors which combinatorially explains certain well known partition congruences mod 5. We give two analogs of Hammond and Lewis's birank statistic. One analog is in terms of…

Bringmann, Mahlburg, and Rhoades have found asymptotic expressions for all moments of the partition statistics rank and crank. In this work we extend their methods to higher ranks. The $T$-rank, introduced by Garvan, for odd integers T=3 is…

Kaur, Rana, and Eyyunni recently defined the mex sequence of a partition and established, by analytic methods, connections to two disparate types of partition-related objects. We make a bijection between partitions with certain mex…

Recently, Andrews and Newman studied the minimal excludant of a partition, which is defined as the smallest positive integer that is not a part of a partition. In this article, we consider the minimal excludant size of an overpartition,…

In this article we study the "norm" of an integer partition, which we define to be the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of…