{"paper":{"title":"The average simplex cardinality of a finite abstract simplicial complex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Oliver Knill","submitted_at":"2019-05-06T16:06:21Z","abstract_excerpt":"We study the average simplex cardinality Dim^+(G) = sum_x |x|/(|G|+1) of a finite abstract simplicial complex G. The functional is a homomorphism from the monoid of simplicial complexes to the rationals: the formula Dim^+(G + H) = Dim^+(G) + Dim^+(H) holds for the join + similarly as for the augmented inductive dimension dim^+(G) = dim(G)+1 where dim is the inductive dimension dim(G) = 1+ sum_x dim(S(x))/|G| with unit sphere S(x) (a recent theorem of Betre and Salinger). In terms of the generating function f(t) = 1+v_0 t + v_1 t^2 + ... +v_d t^(d+1) defined by the f-vector (v_0,v_1, \\dots) of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.02118","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"}