Andrews与Newman的Mex相关分拆函数的奇偶性考虑

Dyson's Crank and the Mex of Integer Partitions

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 parity of the mex has been associated to the crank statistic named by Dyson in 1944. In this note, we extend and strengthen the connection between the crank and mex (along with a new generaliztion of the mex) by proving a number of properties that naturally relate these partition statistics.

A bijective proof of a generalization of the non-negative crank--odd mex identity

Recent works of Andrews--Newman and Hopkins--Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer $n$, there are as many partitions of $n$ with non-negative crank as partitions of $n$ with odd mex. In this paper, we give a bijective proof of a generalization of this identity provided by Hopkins, Sellers, and Stanton. Our method uses an alternative definition of the Durfee decomposition, whose combinatorial link to the crank was recently studied by Hopkins, Sellers, and Yee.

关于r-链最小排除数与最大排除数的组合学

On the combinatorics of r-chain minimal and maximal excludants

本文研究分拆中r-链最小排除数与最大排除数的组合性质，先给出r-chain mex相关定理的组合证明，再引入r-chain maex并用解析与组合两种方法建立其和恒等式。

Mex-related partitions and relations to ordinary partition and singular overpartitions

In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely $p_{t,t}(n)$ and $p_{2t,t}(n)$. We establish identities connecting the ordinary partition function $p(n)$ to $p_{t,t}(n)$ and $p_{2t,t}(n)$ for all $t\geq 1$. Using these identities, we prove that the Ramanujan's famous congruences for $p(n)$ are also satisfied by $p_{t,t}(n)$ and $p_{2t,t}(n)$ for infinitely many values of $t$. Very recently, da Silva and Sellers …

A new generalization of the minimal excludant arising from an analogue of Franklin's identity

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 positive integer $r$. We prove an analogue of Franklin's identity by studying the number of partitions with $j$ multiples of $r$ in total and in the process, discover a natural generalization of the minimal excludant (mex) which we call the $r$-chain mex. Further, we derive the generating function for $σ_{rc} \textup{mex}(n)$, the sum of $r$-chain mex taken ov…

Combinatorial Proof of the Minimal Excludant Theorem

The minimal excludant of a partition $λ$, $\rm{mex}(λ)$, is the smallest positive integer that is not a part of $λ$. For a positive integer $n$, $ σ\, \rm{mex}(n)$ denotes the sum of the minimal excludants of all partitions of $n$. Recently, Andrews and Newman obtained a new combinatorial interpretations for $σ\, \rm{mex}(n)$. They showed, using generating functions, that $σ\, \rm{mex}(n)$ equals the number of partitions of $n$ into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function $σ\, \rm{mex}(n)$. We gen…

Andrews的奇部之下偶部分拆函数及相关无穷乘积的乘法同余

Multiplicative congruences for Andrews's even parts below odd parts function and related infinite products

本文证明了Andrews的奇部之下偶部分拆函数$\overline{\mathcal{EO}}(n)$及相关无穷乘积模$2^{12}$的乘法同余，核心方法结合了Fricke对合与Newman的半整数权Hecke算子技术。

关于与 Ramanujan 的某些 $q$-级数相关的 overpartition 的新最小排除数

On new minimal excludants of overpartitions related to some $q$-series of Ramanujan

受 Andrews 和 Newman 工作启发，本文为 overpartition 定义了四类新的最小排除数，并建立了它们与 Ramanujan 某些函数之间的关系。

关于超划分上奇数最小缺失项之和

On the sum of odd minimal excludants over overpartitions

本文研究超划分上奇数最小缺失项之和，推导其生成函数，建立无限同余族，并给出模4与完全平方数的关系及渐近公式。

Parity distribution and divisibility of Mex-related partition functions

Andrews and Newman introduced the mex-function $\text{mex}_{A,a}(λ)$ for an integer partition $λ$ of a positive integer $n$ as the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $λ$. They then defined $p_{A,a}(n)$ to be the number of partitions $λ$ of $n$ satisfying $\text{mex}_{A,a}(λ)\equiv a\pmod{2A}$. They found the generating function for $p_{t,t}(n)$ and $p_{2t,t}(n)$ for any positive integer $t$, and studied their arithmetic properties for some small values of $t$. In this article, we study the partition function $p_{mt,t}(n)$ for all positive integers $m$ a…

关于分拆的最小 $r$-间隙矩及 Baruah 与 Talukdar 猜想

On the Moments of Least $s$-Gaps of Partitions and a Conjecture of Baruah and Talukdar

本文研究分拆的最小 $r$-间隙的 $k$ 阶矩，给出精确公式，并证明 Baruah 与 Talukdar 关于奇偶最小 $r$-间隙和渐近等价的猜想，同时推广了相关恒等式。

关于分区中缺失整数数量的研究

On the number of missing integers in partitions

本文研究了分区中缺失正整数的整体情况，探讨了给定缺失整数数量的分区数目，并提出了三个关于相关函数的偏差不等式猜想。

Finding Modular Functions for Ramanujan-Type Identities

This paper is concerned with a class of partition functions $a(n)$ introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorithms, we present an algorithm to find Ramanujan-type identities for $a(mn+t)$. While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for $p(11n+6)$ with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions $\overline{p}(5n+2)$ and $\overline{p}(5n+3)$ and Andrews--P…