{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T22:55:17Z","timestamp":1649026517343},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"A di-sk tree is a rooted binary tree whose nodes are labeled by $\\oplus$ or $\\ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. We construct a combinatorial bijection on di-sk trees proving\u00a0 the two quintuples $(\\mathrm{LMAX},\\mathrm{LMIN},\\mathrm{DESB},\\mathsf{iar},\\mathsf{comp})$ and $(\\mathrm{LMAX},\\mathrm{LMIN},\\mathrm{DESB},\\mathsf{comp},\\mathsf{iar})$ have the same distribution over separable permutations. Here for a permutation $\\pi$, $\\mathrm{LMAX}(\\pi)\/\\mathrm{LMIN}(\\pi)$ is the set of values of the left-to-right maxima\/minima of $\\pi$ and $\\mathrm{DESB}(\\pi)$ is the set of descent bottoms of $\\pi$, while $\\mathsf{comp}(\\pi)$ and $\\mathsf{iar}(\\pi)$ are respectively\u00a0 the number of components of $\\pi$ and the length of initial ascending run of $\\pi$.\u00a0
\nInterestingly, our bijection specializes to a bijection on $312$-avoiding permutations, which provides\u00a0 (up to the classical Knuth\u2013Richards bijection)\u00a0an alternative approach to a result of Rubey (2016) that asserts the\u00a0 two triples $(\\mathrm{LMAX},\\mathsf{iar},\\mathsf{comp})$ and $(\\mathrm{LMAX},\\mathsf{comp},\\mathsf{iar})$ are equidistributed\u00a0 on $321$-avoiding permutations. Rubey's result is a symmetric extension of an equidistribution due to Adin\u2013Bagno\u2013Roichman, which implies the class of $321$-avoiding permutations with a prescribed number of components is Schur positive.\u00a0
\nSome equidistribution results for various statistics concerning tree traversal are presented in the end.<\/jats:p>","DOI":"10.37236\/10484","type":"journal-article","created":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T01:48:45Z","timestamp":1639705725000},"source":"Crossref","is-referenced-by-count":0,"title":["A Combinatorial Bijection on di-sk Trees"],"prefix":"10.37236","volume":"28","author":[{"given":"Shishuo","family":"Fu","sequence":"first","affiliation":[]},{"given":"Zhicong","family":"Lin","sequence":"additional","affiliation":[]},{"given":"Yaling","family":"Wang","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2021,12,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p48\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p48\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T01:48:45Z","timestamp":1639705725000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i4p48"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,17]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,10,8]]}},"URL":"https:\/\/doi.org\/10.37236\/10484","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,17]]}}}