{"paper":{"title":"Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.CO","math.RT"],"primary_cat":"math.QA","authors_text":"Ayumu Hoshino, Jun'ichi Shiraishi","submitted_at":"2018-01-30T11:38:40Z","abstract_excerpt":"We present an explicit formula for the transition matrix $\\mathcal{C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{(1^r)}(x\\,|\\,a,-a,c,-c\\,|\\,q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\\mathcal{C}$ enjoy a set of three term recursion relations, which can be regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type $(C_n,C_n)$ Macdonald polynomials $P^{(C_n,C_n)}_{(1^r)}(x\\,|\\,b;q,t)= P_{(1^r)}\\b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09939","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"}