{"paper":{"title":"Bijective recurrences concerning two Schr\\\"oder triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Shishuo Fu, Yaling Wang","submitted_at":"2019-08-11T14:23:44Z","abstract_excerpt":"Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schr\\\"oder paths (resp. little Schr\\\"oder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: \\begin{align*} r(n,0)&=\\sum\\limits_{j=0}^{n-1}2^{j}r(n-1,j), r(n,k)&=r(n-1,k-1)+\\sum\\limits_{j=k}^{n-1}2^{j-k}r(n-1,j),\\quad 1\\le k\\le n, s(n,0) &=\\sum\\limits_{j=1}^{n-1}2\\cdot3^{j-1}s(n-1,j), s(n,k) &=s(n-1,k-1)+\\sum\\limits_{j=k+1}^{n-1}2\\cdot3^{j-k-1}s(n-1,j),\\quad 1\\le k\\le n. \\end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,k\\ge 0}$ and $[s(n,k)]_{n,k\\ge 0}$, who"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03912","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03912/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}