{"paper":{"title":"Enumeration of linear chord diagrams","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"C. M. Reidys, J. E. Andersen, M. S. Waterman, R. C. Penner","submitted_at":"2010-10-27T09:10:59Z","abstract_excerpt":"A linear chord diagram canonically determines a fatgraph and hence has an associated genus $g$. We compute the natural generating function ${\\bf C}_g(z)=\\sum_{n\\geq 0} {\\bf c}_g(n)z^n$ for the number ${\\bf c}_g(n)$ of linear chord diagrams of fixed genus $g\\geq 1$ with a given number $n\\geq 0$ of chords and find the remarkably simple formula ${\\bf C}_g(z)=z^{2g}R_g(z) (1-4z)^{{1\\over 2}-3g}$, where $R_g(z)$ is a polynomial of degree at most $g-1$ with integral coefficients satisfying $R_g({1\\over 4})\\neq 0$ and $R_g(0) = {\\bf c}_g(2g)\\neq 0.$ In particular, ${\\bf C}_g(z)$ is algebraic over $\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.5614","kind":"arxiv","version":1},"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"}