Combinatorial Proof of the Minimal Excludant Theorem

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…

关于超分拆的某些最小排除数的一个注记

A Note On Certain Minimal Excludants Over Overpartitions

本文从算术角度研究超分拆的最小排除数之和及其偶数版本，证明了模2^2的同余式及渐近等价关系，并给出相关渐近行为。

关于超分拆的最小排除数

On Minimal Excludant over Overpartitions

本研究针对超分拆的最小排除数，提出仅考虑划线部分和同时考虑划线与非划线部分两种定义，并探讨相应σ函数的组合、渐近和算术性质。

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。

Overpartitions 上的 Minimal Excludant

Minimal Excludant over Overpartitions

本文定义了 overpartition 上的 minimal excludant，证明了其求和函数等价于三色不同部分分拆数，给出渐近公式，并推广至最小 r-gaps 情形研究其生成函数与算术密度。

关于$k$-色mex相关分拆函数

On $k$-color mex-related partition functions

研究一组$k$-色分拆及其子集与Andrews-Newman最小排除函数的关系，推广了Andrews和Bachraoui的分拆函数并定义了若干新函数。

拓展安德鲁斯和纽曼对 crank-mex 定理的细化

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

本文基于 crank-mex 定理，研究偶数 mex 分拆与负 crank 和正 crank 分拆的关系，通过分析和组合方法证明了相关结果。

$σ_o\ ext{mex}(n)$ 与 $σ_e\ ext{mex}(n)$ 的算术性质及渐近公式

Arithmetic properties and asymptotic formulae for $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$

针对奇、偶最小排除数求和函数，本文研究其模任意 $2$ 的幂的间隙性，证明模 $4$、$8$ 的无限同余式族，并给出 Hardy-Ramanujan 型渐近公式。

Divisibility and distribution of $MEX$ related integer partitions of Andrews and Newman

Andrews and Newman introduced the minimal excludant or ``$mex$'' function for an integer partition $π$ of a positive integer $n$, $mex(π)$, as the smallest positive integer that is not a part of $π$. They defined $σmex(n)$ to be the sum of $mex(π)$ taken over all partitions $π$ of $n$. We prove infinite families of congruence and multiplicative formulas for $σmex(n)$. By restricting to the part of $π$, Andrews and Newman also introduced $moex(π)$ to be the smallest odd integer that is not a part of $π$ and $σmoex(n)$ to be the sum of $moex(π)$ taken over all partitions $π$ of $n$. In this arti…

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。

On Mex-related partition functions of Andrews and Newman

The minimal excludant, or "mex" function, on a set $S$ of positive integers is the least positive integer not in $S$. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely $p_{t,t}(n)$, and provide complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$. In this article, we study the parity of $p_{t,t}(n)$ when $t=2^α, 3\cdot 2^α$ for …

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 …

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。

Combinatorial degenerations of surfaces and Calabi--Yau threefolds

In this article we study combinatorial degenerations of minimal surfaces of Kodaira dimension 0 over local fields, and in particular show that the `type' of the degeneration can be read off from the monodromy operator acting on a suitable cohomology group. This can be viewed as an arithmetic analogue of results of Persson and Kulikov on degenerations of complex surfaces, and extends various particular cases studied by Matsumoto, Liedtke/Matsumoto and Hernández-Mada. We also study `maximally unipotent' degenerations of Calabi--Yau threefolds, following Kollár/Xu, showing in this case that the d…

Nets in groups, minimum length $g$-adic representations, and minimal additive complements

The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,...} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality…