{
  "abstract": "We study the operators obtained by inserting copies of a new largest letter into multiset permutations. Let $G_r$ denote the operator which inserts $r$ copies of a new largest letter. After the change of variables $δ=y-x$, $u=x/y$, and $E=u\\partial_u$, we find that $$G_r=\\frac{δ^r}{r!}E(E+1)\\cdots(E+r-1).$$ Its generating series acts by a rational substitution, which yields the composition law. Our main result gives a common symmetric-function explanation for the ordinary and major-index operators. For $N\\geq 0$, define $Φ_N(F_{N,S})=x^{|S|+1}y^{N-|S|}$. We prove that multiplication by the complete homogeneous symmetric function $h_r$ becomes the ordinary insertion operator: $Φ_{N+r}(h_r f)=G_rΦ_N(f)$, where $f\\in\\mathrm{QSym}_N$. There is a parallel specialization for the major index. A reverse finite principal specialization sends multiplication by $h_r$ to an operator $Q_r$, which is a polynomial in the $q$-shift $Θ_qf(t)=f(qt)$. Thus the ordinary and major-index operators arise from the same multiplication operator $f\\mapsto h_r f$. Since the functions $h_r$ freely generate the ring of symmetric functions, the assignment $h_r\\mapsto G_r$ extends to an algebra homomorphism. We determine the kernel of this homomorphism and the image of every homogeneous component. The images of Schur functions satisfy the Littlewood--Richardson multiplication identities, and the one-row case gives an Eulerian form of the Pieri rule.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03881",
  "author_total": 1,
  "authors": [
    {
      "name": "Shi-Mei Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/938rem0kfjyx3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03881",
  "fulltext_url": null,
  "id": "938rem0kfjyx3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03881",
  "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": "W7208771540",
  "pdf_url": "https://arxiv.org/pdf/2609.03881",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/938rem0kfjyx3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03881",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03881"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Eulerian insertion operators and an Eulerian form of the Pieri rule",
  "title_text": "Eulerian insertion operators and an Eulerian form of the Pieri rule",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/938rem0kfjyx3",
  "venue": "arXiv",
  "versions": []
}