{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T15:48:54Z","timestamp":1759938534125,"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 ${\\cal C}(d,n)$ denote the set of $d$-dimensional lattice  paths using the steps  $X_1 := (1, 0, \\ldots, 0),$ $ X_2 := (0, 1, \\ldots, 0),$  $\\ldots,$ $ X_d := (0,0, \\ldots,1)$, running from $(0,\\ldots,0)$  to $(n,\\ldots,n)$,  and lying in  $\\{(x_1,x_2, \\ldots, x_d) :  0 \\le x_1 \\le x_2 \\le  \\ldots \\le  x_d \\}$. On any  path $P:=p_1p_2 \\ldots p_{dn} \\in {\\cal C}(d,n)$,  define the statistics ${\\rm asc}(P) := $$|\\{i : p_ip_{i+1} = X_jX_{\\ell}, j &lt; \\ell  \\}|$ and ${\\rm des}(P) := $$|\\{i : p_ip_{i+1} = X_jX_{\\ell}, j&gt;\\ell  \\}|$. Define the generalized Narayana number $N(d,n,k)$ to count the paths in ${\\cal C}(d,n)$ with ${\\rm asc}(P)=k$.  We consider the derivation of a formula for $N(d,n,k)$, implicit in MacMahon's work.  We examine other statistics for $N(d,n,k)$ and show that the statistics ${\\rm asc}$ and ${\\rm des}-d+1$ are equidistributed.  We use Wegschaider's algorithm, extending Sister Celine's (Wilf-Zeilberger) method to multiple summation, to obtain recurrences for $N(3,n,k)$.  We introduce the generalized large Schr\u00f6der numbers $(2^{d-1}\\sum_k N(d,n,k)2^k)_{n\\ge1}$ to count constrained paths using step sets which include diagonal steps.<\/jats:p>","DOI":"10.37236\/1807","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:32:34Z","timestamp":1578691954000},"source":"Crossref","is-referenced-by-count":14,"title":["Generalizing Narayana and Schr\u00f6der Numbers to Higher Dimensions"],"prefix":"10.37236","volume":"11","author":[{"given":"Robert A.","family":"Sulanke","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,8,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r54\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r54\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:01:32Z","timestamp":1579305692000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r54"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,8,23]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1807","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,8,23]]},"article-number":"R54"}}