{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T20:02:42Z","timestamp":1760385762209,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We notice that two combinatorial interpretations of the well-known Catalan numbers $C_n=(2n)!\/n!(n+1)!$ naturally give rise to a recursion for $C_n$. This recursion is ideal for the study of the congruences of $C_n$ modulo $2^r$, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh's recent classification of $C_n$ modulo $2^r$. The equivalence $C_{n}\\equiv_{2^r} C_{\\bar n}$ is further reduced to $C_{n}\\equiv_{2^r} C_{\\tilde{n}}$ for simpler $\\tilde{n}$. Moreover, by using connections between weighted Dyck paths and Motzkin paths, we find new classes of combinatorial sequences whose $2$-adic order is equal to that of $C_n$, which is one less than the sum of the digits of the binary expansion of $n+1$.<\/jats:p>","DOI":"10.37236\/664","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:42:35Z","timestamp":1578714155000},"source":"Crossref","is-referenced-by-count":2,"title":["A Short Approach to Catalan Numbers Modulo $2^r$"],"prefix":"10.37236","volume":"18","author":[{"given":"Guoce","family":"Xin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing-Feng","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,9,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p177\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p177\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:02:17Z","timestamp":1579302137000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p177"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,9,9]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/664","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,9,9]]},"article-number":"P177"}}