{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:24:15Z","timestamp":1759335855719,"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>The symmetric $q,t$-Catalan polynomial $C_n(q,t)$, which specializes to the Catalan polynomial $C_n(q)$ when $t=1$, was defined by Garsia and Haiman in 1994.  In 2000, Garsia and Haglund described statistics $a(\\pi)$ and $b(\\pi)$ on Dyck paths such that $C_n(q,t) = \\sum_{\\pi} q^{a(\\pi)}t^{b(\\pi)}$ where the sum is over all $n \\times n$ Dyck paths.  Specializing $t=1$ gives the Catalan polynomial $C_n(q)$ defined by Carlitz and Riordan and further studied by Carlitz.   Specializing both $t=1$ and $q=1$ gives the usual Catalan number $C_n$.  The Catalan number $C_n$ is known to count the number of $n \\times n$ Dyck paths and the number of $312$-avoiding permutations in $S_n$, as well as at least 64 other combinatorial objects.  In this paper, we define a bijection between Dyck paths and $312$-avoiding permutations which takes the area statistic $a(\\pi)$ on Dyck paths to the inversion statistic on $312$-avoiding permutations. The inversion statistic can be thought of as the number of $(21)$ patterns in a permutation $\\sigma$.  We give a characterization for the number of $(321)$, $(4321)$, $\\dots$, $(k\\cdots21)$ patterns that occur in $\\sigma$ in terms of the corresponding Dyck path.<\/jats:p>","DOI":"10.37236\/1584","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:06:36Z","timestamp":1578708396000},"source":"Crossref","is-referenced-by-count":5,"title":["An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding Permutations"],"prefix":"10.37236","volume":"8","author":[{"given":"Jason","family":"Bandlow","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kendra","family":"Killpatrick","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2001,12,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r40\/comment","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r40\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:16:20Z","timestamp":1579324580000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v8i1r40"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,12,10]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2001,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1584","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2001,12,10]]},"article-number":"R40"}}