{"paper":{"title":"Betti Numbers of Cut Complexes of Squared Paths and a Recurrence Conjecture","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yaoran Yang, Yutong Zhang","submitted_at":"2026-05-21T17:57:09Z","abstract_excerpt":"For a graph $G$ on $[n]$, the $k$-cut complex $\\Delta_k(G)$ has facets $[n]\\setminus T$, where $T$ ranges over the disconnected $k$-vertex induced subgraphs of $G$. Bayer, Denker, Jeli\\'c Milutinovi\\'c, Sundaram, and Xue proved that the $k$-cut complex of the squared path $P_n^2$ is shellable for $n\\ge k+3$ and conjectured a finite-difference recurrence for its top reduced Betti number along every diagonal $n-k=r$. We prove the recurrence by giving the exact formula $\\beta(k,n)=\\binom{n-1}{k-1}-\\sum_{j=0}^{\\min\\{k-1,n-k\\}}\\binom{k-1}{j}(n-k-j+1)+(n-k)$ for $r=n-k\\ge3$. Equivalently, for fixed "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22808","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.22808/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}