随机Carlitz组合中不同零件尺寸的平均数

随机整数组合中不同零件尺寸的预期数量

Expected number of distinct part sizes in a random integer composition

得到了整数n的随机组合中不同零件尺寸的期望个数的渐近性，即$n\to\infty$。

紧马尔可夫链与随机合成

Tight Markov chains and random compositions

对于$\Bbb N$上的遍历马尔可夫链$\{X(t)\}$，具有平稳分布$π$，令$T_n>0$表示$[n]^c$的命中时间，令$X_n=X(T_n)$。在2005年左右，Guy Louchard推广了一个猜想，即对于$n\to \infty$，$T_n$几乎是几何的（$p$），$p=π([n]^c)$，$X_n$几乎是平稳分布在$[n]^c$上，并且如果$p(n):=\sup_ip(i,[n]^c)\to 0$以指数速度增长，则$X_n$和$T_n$几乎是独立的。对于链与$p(n) \to 0$然而缓慢，并与$\sup_{i,j}\,\|p(i,\cdot)-p(j,\cdot)\|_{TV}<1$，我们表明，Louchard的猜想确实是真实的，即使是命中的任意$S_n\subset\Bbb N$与$π(S_n)\to 0$。更确切地说，在$k\,\sup_ip(i,S_n)$阶的总变化距离内，通过$(\ell_n,t_n)$的独立拷贝的$k$长序列来近似与自先前命中（对于第一命中，自起始时刻起）起经过的时间配对的$k$连续命中位置的序列，其中$\ell_n= \text{Geometric}\,(π(S_n))$、$t_n$静止地分布在$S_n$上，并且$\ell_n$独立于$t_n$。这两个条件很容易满足的马尔可夫链，出现在卢查德的研究作为可能尖锐的近似两个随机组成的一个大整…

整数分拆的迭代不同绝对差

Iterated Distinct Absolute Differences of Integer Compositions

该研究确定了具有指定深度的整数分拆对应的最小正整数，推导了其公式，证明了≥该值的整数均有对应深度分拆，还对达最小a(n)的分拆进行分类计数并给出公式。

On The Composition Lemma

We describe two distinct simple, short and self contained proofs of the composition lemma.

Shift-plethysm, Hydra continued fractions, and m-distinct partitions

We introduce the hydra continued fractions, as a generalization of the Rogers-Ramanujan continued fractions, and give a combinatorial interpretation in terms of shift-plethystic trees. We then show it is possible to express them as a quotient of m-distinct partition generating functions, and in its dual form as a quotient of the generating functions of compositions with contiguous rises upper bounded by m-1. We obtain new generating functions for compositions according to their local minima, for partitions with a prescribed set of rises, and for compositions with prescribed sets of contiguous …

具有受限部分数的偶与奇组合

Even and odd compositions with restricted parts

本文将Legendre关于奇偶划分差值的经典结果推广到组合情形，并建立了与OEIS中多个序列条目相关的若干推广。

On the number of simultaneous core partitions with $d$-distinct parts

We investigate the number $N_{d,r}(s)$ of $(s, s+r)$-core integer partitions with $d$-distinct parts. Our first main result is a proof of a recurrence relation conjectured by Sahin in 2018. We also derive generating functions, asymptotics, and exact formulas for $N_{d,r}(s)$ when $r$ is within $d$ of a multiple of $s$. Finally, we exhibit a surprising connection to $A$-restricted compositions.

Compositions with restricted parts

Euler showed that the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on partitions with restricted parts, which were confirmed analytically by Andrews and Chern and combinatorially by Yang. Analogous to Euler's partition theorem, it is known that the number of compositions of $n$ with odd parts equals the number of compositions of $n+1$ with parts greater than one, as both numbers equal the Fibonacci number $F_n$. Recently, Sills provid…

Kadell的正交性猜想的推广

A generalization of Kadell's orthogonality ex-conjecture

本文基于Kadell的正交性猜想，通过将变量分为两部分，推广了Zhou在2021年对任意弱组合v的恒等式递推关系的研究结果。