{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:26:04Z","timestamp":1759335964763,"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>Restricted growth functions (RGFs) avoiding the pattern $1212$ are in natural bijection with noncrossing partitions. Motivated by recent work of Campbell et al., we study five classical statistics bk, ls, lb, rs and rb\u00a0on $1212$-avoiding RGFs. We show the equidistribution of (ls,\u00a0rb, lb, bk) and (rb, ls, lb, bk) on $1212$-avoiding RGFs by constructing a simple involution.\u00a0To our surprise, this result was already proved by Simion 22 years ago via an involution on noncrossing partitions. Our involution, though turns out essentially the same as Simion's, is defined quite differently and has the advantage that makes the discussion more transparent. Consequently, a multiset-valued extension of Simion's result is discovered. Furthermore, similar approach enables us to prove the equidistribution of (mak,\u00a0rb, rs, bk) and (rb, mak, rs, bk) on $1212$-avoiding RGFs, where \"mak\" is a set partition statistic introduced by Steingr\u00edmsson.Through two bijections to Motzkin paths, we also prove that the triple of classical permutation statistics (exc+1, den, inv\u00a0\u2014\u00a0exc) on $321$-avoiding permutations is equidistributed with the triple (bk, rb, rs) on $1212$-avoiding RGFs, which generalizes another result of Simion. In the course, an interesting $q$-analog of the $\\gamma$-positivity of Narayana polynomials is found.<\/jats:p>","DOI":"10.37236\/6728","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:59:12Z","timestamp":1578671952000},"source":"Crossref","is-referenced-by-count":4,"title":["On 1212-Avoiding Restricted Growth Functions"],"prefix":"10.37236","volume":"24","author":[{"given":"Zhicong","family":"Lin","sequence":"first","affiliation":[]},{"given":"Shishuo","family":"Fu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,3,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p53\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p53\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:03:18Z","timestamp":1579237398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i1p53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1,20]]}},"URL":"https:\/\/doi.org\/10.37236\/6728","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,3,17]]},"article-number":"P1.53"}}