{"paper":{"title":"Combinatorial interpretations of Tutte polynomials at the point $(2,-1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tianlong Ma, Xian'an Jin, Yu Chen","submitted_at":"2026-05-30T09:55:58Z","abstract_excerpt":"Let $G$ be a simple connected graph, and let $T_{G}(x,y)$ be the Tutte polynomial of $G$. Motivated by the works in \\cite{Ma}, we, in this paper, introduce the even-left spanning forests of $G$ and odd $G$-partitionable permutations, and show that $T_{G}(2,-1)$ is equal to both the number of even-left spanning forests of $G$ and the number of odd $G$-partitionable permutations. In particular, for a complete graph $K_n$, we prove that $T_{K_{n}}(2,-1)$ is the number of alternating permutations on $\\{1,2,\\dots,n+1\\}$, using two distinct techniques: a recurrence relation and an explicit bijection"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00653","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/2606.00653/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"}