{"paper":{"title":"Line configurations and r-Stirling partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Timothy Wilson, Brendon Rhoades","submitted_at":"2018-04-21T02:28:56Z","abstract_excerpt":"A set partition of $[n] := \\{1, 2, \\dots, n \\}$ is called {\\em $r$-Stirling} if the numbers $1, 2, \\dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k \\leq n$ whose algebraic properties are governed by the combinatorics of ordered set partitions of $[n]$ with $k$ blocks. We introduce a variant $R_{n,k}^{(r)}$ of this quotient for ordered $r$-Stirling partitions which depends on three integers $r \\leq k \\leq n$. We describe the standard monomial basis of $R_{n,k}^{(r)}$ and use the combinatorial notion of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.07879","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"}