{
  "abstract": "We study three closely related questions arising from the recent work of Andrews and El Bachraoui on the two-color partition series \\[ S_1(q)=\\sum_{n\\ge0}s_1(n)q^n=\\sum_{a\\ge0}q^a(-q^{a+1};q)_\\infty^2 \\] and its odd companion, denoted by $T_o(q)$. First, for the eta-normalized companion \\[ C(q)=(q;q)_\\infty T_o(q)=\\sum_{n\\ge0}c(n)q^n, \\] we prove a strong form of the Andrews--El Bachraoui sign conjecture that $\\limsup c(n)=+\\infty$ and $\\liminf c(n)=-\\infty$. Second, we construct an involution using the Franklin-type involution of Chen and Liu to combinatorially explain Andrews--El Bachraoui congruence for $s_1(n)$ modulo 4. Third, we introduce a two-variable refinement of $S_1(q)$ which records the difference of the lengths of the two colored components. Using Ramanujan's partial theta identity and heat-type equations, we obtain a closed formula for all of its even moments, and we show that the eta-normalized moments satisfy termwise congruences modulo $p^{2k}$ for every prime $p\\equiv5\\pmod6$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.10576",
  "author_total": 3,
  "authors": [
    {
      "name": "Aritram Dhar"
    },
    {
      "name": "Ankush Goswami"
    },
    {
      "name": "Mohit Tripathi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8e7gyghzmhgkc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.10576",
  "fulltext_url": null,
  "id": "8e7gyghzmhgkc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.10576",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7168259003",
  "pdf_url": "https://arxiv.org/pdf/2607.10576",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8e7gyghzmhgkc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2607.10576",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.10576"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sign Patterns in a Two Colored Partition Companion series",
  "title_text": "Sign Patterns in a Two Colored Partition Companion series",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/8e7gyghzmhgkc",
  "venue": "arXiv",
  "versions": []
}