{"paper":{"title":"On The Morse Ensemble Polynomial Of Simplicial Complexes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT","math.SP"],"primary_cat":"math.CO","authors_text":"Chong Zheng","submitted_at":"2026-05-23T17:56:04Z","abstract_excerpt":"We introduce the \\emph{Morse ensemble polynomial} $\\ME_K(z_0,\\ldots,z_d)$ of a finite simplicial complex $K$, defined as the generating function $\\ME_K = \\sum_M \\prod_i z_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$, where $c_i(M)$ counts critical $i$-simplices. This polynomial records the complete distribution of Morse vectors across all discrete Morse functions on $K$, and is an isomorphism invariant of simplicial complexes.\n  Our main results are the following. \\textbf{(I) The Laplacian Formula}: for any connected graph $G$, $\\ME_G = z_1^{m-n}\\det(z_0z_1\\,I_n + L_G)$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24689","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/2605.24689/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"}