{"paper":{"title":"The complexity of the fermionant, and immanants of constant width","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.str-el","math.CO"],"primary_cat":"cs.CC","authors_text":"Cristopher Moore, Stephan Mertens","submitted_at":"2011-10-09T11:16:48Z","abstract_excerpt":"In the context of statistical physics, Chandrasekharan and Wiese recently introduced the \\emph{fermionant} $\\Ferm_k$, a determinant-like quantity where each permutation $\\pi$ is weighted by $-k$ raised to the number of cycles in $\\pi$. We show that computing $\\Ferm_k$ is #P-hard under Turing reductions for any constant $k > 2$, and is $\\oplusP$-hard for $k=2$, even for the adjacency matrices of planar graphs. As a consequence, unless the polynomial hierarchy collapses, it is impossible to compute the immanant $\\Imm_\\lambda \\,A$ as a function of the Young diagram $\\lambda$ in polynomial time, e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1821","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"}