Longest Run of Equal Parts in a Random Integer Composition

The distribution of longest run lengths in integer compositions

We find the generating function for $C(n,k,r)$, the number of compositions of $n$ into $k$ positive parts all of whose runs (contiguous blocks of constant parts) have lengths less than $r$, using recent generalizations of the method of Guibas and Odlyzko for finding the number of words that avoid a given list of subwords.

关于随机整数拆分序列的演化

On the evolution of random integer compositions

本文研究了 $m$ 从零增加时随机 $n$ 项弱整数拆分序列渐近结构的演化，确立了多种子结构（如零/非零项游程、递增游程、等项游程、最大正方形等模式）出现与消失的阈值，并揭示了精确连续模式出现与消失的二分性规律。

多元随机序列的第 m 长游程

The m-th Longest Runs of Multivariate Random Sequences

本文针对由 k 种字母组成的多元随机序列，提出两种游程长度排序定义，并利用两步方法推导出第 m 长游程分布的精确公式，该方法还可推广到其他游程相关分布。

Variance of Longest Run Duration in a Random Bitstring

We continue an earlier study, starting with unconstrained $n$-bitstrings, focusing now less on average behavior and more on uncertainty. The interplay between $\bullet$ longest runs of 0s and of 1s, when bitstrings are multus $\bullet$ longest runs of 0s and bitsums (# of 1s), when bitstrings are solus $\\$ is examined. While negative correlations approach zero as $n \rightarrow \infty$ in the former (for clumped 1s), the limit is evidently nonzero in the latter (for separated 1s). Similar analysis is possible when both 0s and 1s are clumped (bimultus), and when 0s are clumped but 1s are separ…

Run and Frequency quotas for q-binary trials

We study the distributions of waiting times in variations of the $q$-sooner and later waiting time problem. One variation imposes length and frequency quotas on the runs of successes and failures. Another case considers binary trials for which the probability of ones is geometrically varying. We also study the distributions of Longest run under the same variations. The main theorems are sooner and later waiting time problem and the joint distribution of the length of the longest success and longest failure runs when a run and frequency quotas imposed on runs of successes and failure. In the pr…

Locally Restricted Sequential Structures and Runs of a Subcomposition in Integer Compositions

We study part sizes of supercritical locally restricted sequential structures. This extends previous results about locally restricted integer compositions and part sizes in smooth supercritical compositional structures. Applications are given for runs of subcompositions. The problems are formulated as enumerating directed walks in sized infinite digraphs and the proofs depend heavily on earlier results by Bender and Canfield about infinite transfer matrices.