{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:29Z","timestamp":1753893869850,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We will study the inversion statistic of $321$-avoiding permutations, and obtain that the number of $321$-avoiding permutations on $[n]$ with $m$ inversions is given by\\[|\\mathcal {S}_{n,m}(321)|=\\sum_{b \\vdash m}{n-\\frac{\\Delta(b)}{2}\\choose l(b)}.\\]where the sum runs over all compositions\u00a0$b=(b_1,b_2,\\ldots,b_k)$ of $m$, i.e.,\\[m=b_1+b_2+\\cdots+b_k \\quad{\\rm and}\\quad \u00a0b_i\\ge 1,\\]$l(b)=k$ is the length of $b$, and $\\Delta(b):=|b_1|+|b_2-b_1|+\\cdots+|b_k-b_{k-1}|+|b_k|$.\u00a0We obtain a new bijection from $321$-avoiding permutations to Dyck paths which establishes a relation on inversion number of $321$-avoiding permutations and valley height of Dyck paths.<\/jats:p>","DOI":"10.37236\/5451","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:39:42Z","timestamp":1578695982000},"source":"Crossref","is-referenced-by-count":1,"title":["Inversion Formulae on Permutations Avoiding 321"],"prefix":"10.37236","volume":"22","author":[{"given":"Pingge","family":"Chen","sequence":"first","affiliation":[]},{"given":"Zhousheng","family":"Mei","sequence":"additional","affiliation":[]},{"given":"Suijie","family":"Wang","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,11,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i4p28\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i4p28\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:08:10Z","timestamp":1579255690000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i4p28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,11,13]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,10,16]]}},"URL":"https:\/\/doi.org\/10.37236\/5451","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,11,13]]},"article-number":"P4.28"}}