{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:59Z","timestamp":1753893839490,"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>Let $\\Phi$ denote\u00a0 Foata's second fundamental transformation\u00a0on permutations.\u00a0For a permutation $\\sigma$ in the symmetric group $S_n$, let\u00a0$\\widetilde{\\Lambda}_{\\sigma}=\\{\\pi\\in S_n\\colon\\pi\\leq_{w} \\sigma\\}$ be the\u00a0principal order ideal generated by $\\sigma$\u00a0\u00a0in the weak order $\\leq_{w}$.\u00a0Bj\u00f6rner and Wachs have shown that $\\widetilde{\\Lambda}_{\\sigma}$ is\u00a0invariant under $\\Phi$\u00a0if and only if $\\sigma$ is a 132-avoiding permutation.\u00a0In this paper, we consider the invariance property of\u00a0 $\\Phi$ on the principal order ideals ${\\Lambda}_{\\sigma}=\\{\\pi\\in S_n\\colon \\pi\\leq \\sigma\\}$\u00a0with respect to the Bruhat order $\\leq$.\u00a0 We obtain\u00a0a characterization\u00a0 of permutations $\\sigma$ such that\u00a0${\\Lambda}_{\\sigma}$ are invariant under $\\Phi$.\u00a0We also consider the invariant principal order \u00a0ideals with respect to the Bruhat order\u00a0\u00a0under Han's bijection $H$. We find\u00a0\u00a0that ${\\Lambda}_{\\sigma}$ is invariant under\u00a0the bijection $H$ if and only if\u00a0it is invariant under the transformation $\\Phi$.<\/jats:p>","DOI":"10.37236\/2680","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:12:35Z","timestamp":1578712355000},"source":"Crossref","is-referenced-by-count":1,"title":["Invariant Principal Order Ideals under Foata\u2019s Transformation"],"prefix":"10.37236","volume":"19","author":[{"given":"Teresa X.S.","family":"Li","sequence":"first","affiliation":[]},{"given":"Melissa Y.F.","family":"Miao","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2012,10,18]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i4p3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i4p3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:28:58Z","timestamp":1579300138000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i4p3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,18]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2012,10,18]]}},"URL":"https:\/\/doi.org\/10.37236\/2680","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,10,18]]},"article-number":"P3"}}