{"paper":{"title":"On symmetries in phylogenetic trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"\\'Eric Fusy","submitted_at":"2016-02-24T08:48:07Z","abstract_excerpt":"Billey et al. [arXiv:1507.04976] have recently discovered a surprisingly simple formula for the number $a_n(\\sigma)$ of leaf-labelled rooted non-embedded binary trees (also known as phylogenetic trees) with $n\\geq 1$ leaves, fixed (for the relabelling action) by a given permutation $\\sigma\\in\\frak{S}_n$. Denoting by $\\lambda\\vdash n$ the integer partition giving the sizes of the cycles of $\\sigma$ in non-increasing order, they show by a guessing/checking approach that if $\\lambda$ is a binary partition (it is known that $a_n(\\sigma)=0$ otherwise), then $$ a_n(\\sigma)=\\prod_{i=2}^{\\ell(\\lambda)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.07432","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}