{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:42:57Z","timestamp":1753893777207,"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>Given a barrier $0 \\leq b_0 \\leq b_1 \\leq \\cdots$, let $f(n)$ be the number of nondecreasing integer sequences $0 \\leq a_0 \\leq a_1 \\leq \\cdots \\leq a_n$ for which $a_j \\leq b_j$ for all $0 \\leq j \\leq n$. Known formul\u00e6 for $f(n)$ include an $n \\times n$ determinant whose entries are binomial coefficients (Kreweras, 1965) and, in the special case of $b_j = rj+s$, a short explicit formula (Proctor, 1988, p.320). A relatively easy bivariate recursion, decomposing all sequences according to $n$ and $a_n$, leads to a bivariate generating function, then a univariate generating function, then a linear recursion for $\\{ f(n) \\}$.  Moreover, the coefficients of the bivariate generating function have a probabilistic interpretation, leading to an analytic inequality which is an identity for certain values of its argument.<\/jats:p>","DOI":"10.37236\/149","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:31:40Z","timestamp":1578717100000},"source":"Crossref","is-referenced-by-count":2,"title":["Counting Nondecreasing Integer Sequences that Lie Below a Barrier"],"prefix":"10.37236","volume":"16","author":[{"given":"Robin","family":"Pemantle","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Herbert S.","family":"Wilf","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2009,5,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r60\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r60\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:55:27Z","timestamp":1579316127000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r60"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,5,11]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/149","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,5,11]]},"article-number":"R60"}}