{
  "id": "2412.03086",
  "version": "v1",
  "published": "2024-12-04T07:27:31.000Z",
  "updated": "2024-12-04T07:27:31.000Z",
  "title": "Complete homogeneous symmetric polynomials with repeating variables",
  "authors": [
    "Luis Angel González-Serrano",
    "Egor A. Maximenko"
  ],
  "comment": "28 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider polynomials of the form $\\operatorname{h}_m(y_1^{[\\varkappa_1]},\\ldots,y_n^{[\\varkappa_n]})$, where $\\operatorname{h}_m$ is the complete homogeneous polynomial of degree $m$ and $y_j^{[\\varkappa_j]}$ denotes $y_j$ repeated $\\varkappa_j$ times. Using the decomposition of the generating function into partial fractions we represent such polynomials in the form \\[ \\operatorname{h}_m(y_1^{[\\varkappa_1]},\\ldots,y_n^{[\\varkappa_n]}) =\\sum_{j=1}^n \\sum_{r=1}^{\\varkappa_j} \\binom{r+m-1}{r-1} A_{y,\\varkappa,j,r} y_j^m, \\] where $A_{y,\\varkappa,j,r}$ are some coefficients that do not depend on $m$. We also provide an alternative proof using the inverse of the confluent Vandermonde matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-04T07:27:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "15A15"
    ],
    "keywords": [
      "complete homogeneous symmetric polynomials",
      "repeating variables",
      "confluent vandermonde matrix",
      "complete homogeneous polynomial",
      "partial fractions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}