{"paper":{"title":"Lattice Polynomials, 12312-Avoiding Partial Matchings and Even Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Louis W. Shapiro, Susan Y. J. Wu, William Y. C. Chen","submitted_at":"2010-11-16T11:12:13Z","abstract_excerpt":"The lattice polynomials $L_{i,j}(x)$ are introduced by Hough and Shapiro as a weighted count of certain lattice paths from the origin to the point $(i,j)$. In particular, $L_{2n, n}(x)$ reduces to the generating function of the numbers $T_{n,k}={1\\over n}{n-1+k\\choose n-1}{2n-k\\choose n+1}$, which can be viewed as a refinement of the $3$-Catalan numbers $T_n=\\frac{1}{2n+1}{3n\\choose n}$. In this paper, we establish a correspondence between $12312$-avoiding partial matchings and lattice paths, and we show that the weighted count of such partial matchings with respect to the number of crossings "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.3650","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"}