{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:40Z","timestamp":1753893820819,"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>Let $a_n$ count the number of $2$-dimensional rook paths $\\mathcal{R}_{n,n}$ from $(0,0)$ to $(2n,0)$. Rook paths $\\mathcal{R}_{m,n}$ are the lattice paths from $(0,0)$ to $(m+n,m-n)$ with allowed steps $(x,x)$ and $(y,-y)$ where $x,y\\in\\mathbb{N}^{+}$. In answer to the open question proposed by M. Erickson et al. (2010), we shall present a combinatorial proof for the recurrence of $a_n$, i.e., $(n+1)a_{n+1}+9(n-1)a_{n-1}=2(5n+2)a_n$ with initial conditions $a_0=1$ and $a_1=2$. Furthermore, our proof can be extended to show the recurrence for the number of multiple Dyck paths $d_n$, i.e., $(n+2)d_{n+1}+9(n-1)d_{n-1}=5(2n+1)d_n$ with $d_0=1$ and $d_1=1$, where $d_n=\\mathcal{N}_n(4)$ and $\\mathcal{N}_n(x)$ is Narayana polynomial.<\/jats:p>","DOI":"10.37236\/2105","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:30:56Z","timestamp":1578713456000},"source":"Crossref","is-referenced-by-count":0,"title":["A Combinatorial Proof of the Recurrence for Rook Paths"],"prefix":"10.37236","volume":"19","author":[{"given":"Emma Yu","family":"Jin","sequence":"first","affiliation":[]},{"given":"Markus E.","family":"Nebel","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2012,3,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p57\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p57\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:40:59Z","timestamp":1579300859000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i1p57"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3,19]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.37236\/2105","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,3,19]]},"article-number":"P57"}}