{
  "id": "2305.19830",
  "version": "v1",
  "published": "2023-05-31T13:14:06.000Z",
  "updated": "2023-05-31T13:14:06.000Z",
  "title": "A family of Counterexamples on Inequality among Symmetric Functions",
  "authors": [
    "Jia Xu",
    "Yong Yao"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO",
    "cs.SC"
  ],
  "abstract": "Inequalities among symmetric functions are fundamental questions in mathematics and have various applications in science and engineering. In this paper, we tackle a conjecture about inequalities among the complete homogeneous symmetric function $H_{n,\\lambda}$, that is, the inequality $H_{n,\\lambda}\\leq H_{n,\\mu}$ implies majorization order $\\lambda\\preceq\\mu$. This conjecture was proposed by Cuttler, Greene and Skandera in 2011. The conjecture is a close analogy with other known results on Muirhead-type inequalities. In 2021, Heaton and Shankar disproved the conjecture by showing a counterexample for degree $d=8$ and number of variables $n=3$. They then asked whether the conjecture is true when~ the number of variables, $n$, is large enough? In this paper, we answer the question by proving that the conjecture does not hold when $d\\geq8$ and $n\\geq2$. A crucial step of the proof relies on variables reduction. Inspired by this, we propose a new conjecture for $H_{n,\\lambda}\\leq H_{n,\\mu}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-31T13:14:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "14P99",
      "90C22"
    ],
    "keywords": [
      "inequality",
      "conjecture",
      "counterexample",
      "implies majorization order",
      "complete homogeneous symmetric function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}