{"paper":{"title":"Linear equations for the number of intervals which are isomorphic with Boolean lattices and the Dehn--Sommerville equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"G\\'abor Heged\\\"us","submitted_at":"2010-02-10T09:44:54Z","abstract_excerpt":"Let $P$ be a finite poset. Let $L:=J(P)$ denote the lattice of order ideals of $P$. Let $b_i(L)$ denote the number of Boolean intervals of $L$ of rank $i$. We construct a simple graph $G(P)$ from our poset $P$. Denote by $f_i(P)$ the number of the cliques $K_{i+1}$, contained in the graph $G(P)$. Our main results are some linear equations connecting the numbers $f_i(P)$ and $b_i(L)$. We reprove the Dehn--Sommerville equations for simplicial polytopes. In our proof we use free resolutions and the theory of Stanley--Reisner rings."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.2049","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"}