{
  "abstract": "Let $σ_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. We completely determine the sensitivity, average sensitivity, and block sensitivity of $σ_{n,d}$ for every $1\\le d\\le n$. Using Lucas' theorem, we obtain a uniform binary description of the Hamming-weight value sequence, from which the sensitivity and average-sensitivity formulas follow and the computation of block sensitivity reduces to at most four explicit candidates. Combining these results with the arbitrary-degree formula for certificate complexity, we determine the exact relations among sensitivity, block sensitivity, and certificate complexity. We also prove a general result for symmetric Boolean functions: every nonconstant symmetric Boolean function $f$ satisfies \\[ \\bs(f)\\le \\max\\{s(f),C(f)-1\\}. \\] Consequently, only the three patterns \\[ s=\\bs=C,\\qquad s=\\bs<C,\\qquad s<\\bs<C \\] can occur for nonconstant symmetric Boolean functions. For elementary symmetric Boolean functions, we give necessary and sufficient conditions for each of these three patterns, thereby completely classifying the relations among $s(σ_{n,d})$, $\\bs(σ_{n,d})$, and $C(σ_{n,d})$. In particular, we obtain a necessary and sufficient characterization of the full strict hierarchy \\[ s(σ_{n,d})<\\bs(σ_{n,d})<C(σ_{n,d}), \\] and exhibit infinite families for which it holds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23781",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuan Li"
    },
    {
      "name": "Jing Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e13aw4r5ptwfd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23781",
  "fulltext_url": null,
  "id": "e13aw4r5ptwfd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23781",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.23781",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e13aw4r5ptwfd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23781",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23781"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sensitivity and Block Sensitivity of Elementary Symmetric Boolean Functions of Arbitrary Degree",
  "title_text": "Sensitivity and Block Sensitivity of Elementary Symmetric Boolean Functions of Arbitrary Degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/e13aw4r5ptwfd",
  "venue": "arXiv",
  "versions": []
}