{
  "abstract": "We study the initial run of equal adjacent parts in an integer composition. For a composition alpha, let P(alpha) be the common value of the parts in the initial run, let K(alpha) be the number of parts in that run, and set S(alpha)=P(alpha)K(alpha). We obtain exact finite formulas for the joint parameters (P,K,S) and the corresponding limiting laws. In the unrestricted model, each fixed event (P,K)=(p,k) reaches its limiting probability once n is above an explicit threshold. The limiting distribution of the total size is P(S=s)=2^{-s} sum_{d|s}(1-2^{-d}). Pointing a unit cell in the initial run gives M_1(z)=((1-z)/(1-2z)) sum_{m>=1} sigma_1(m) z^m. For every r>=1, the r-th power-moment generating function has coefficient function A_r(n)=sum_{j=1}^r (-1)^{j+1} binom(r,j) n^{r-j} sigma_j(n), so higher moments involve explicit finite combinations of generalized divisor sums. We also give factorial-moment formulas for the initial-run length and extend the construction to compositions with parts in a prescribed set A, where the corresponding formulas involve restricted divisor sums and the dominant root of the part generating function. Copy",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30314",
  "author_total": 1,
  "authors": [
    {
      "name": "Igor Kleiner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5efa7zcrn397j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30314",
  "fulltext_url": null,
  "id": "5efa7zcrn397j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30314",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214605038",
  "pdf_url": "https://arxiv.org/pdf/2609.30314",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5efa7zcrn397j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30314",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30314"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Initial Runs in Integer Compositions: Lambert Series and Generalized Divisor Sums",
  "title_text": "Initial Runs in Integer Compositions: Lambert Series and Generalized Divisor Sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/5efa7zcrn397j",
  "venue": "arXiv",
  "versions": []
}