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	<title>Invariant Principal Order Ideals under Foata’s Transformation | Li | The Electronic Journal of Combinatorics</title>
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			<meta name="keywords" content="Foata’s second fundamental transformation; Han’s bijection; Bruhat order; principal order ideal" />
	
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	<meta name="DC.Contributor.Sponsor" xml:lang="en" content=""/>
	<meta name="DC.Creator.PersonalName" content="Teresa X.S. Li"/>
	<meta name="DC.Creator.PersonalName" content="Melissa Y.F. Miao"/>
	<meta name="DC.Date.created" scheme="ISO8601" content="2012-10-18"/>
	<meta name="DC.Date.dateSubmitted" scheme="ISO8601" content="2012-09-01"/>
	<meta name="DC.Date.issued" scheme="ISO8601" content="2012-10-18"/>
	<meta name="DC.Date.modified" scheme="ISO8601" content="2012-10-18"/>
	<meta name="DC.Description" xml:lang="en" content="Let $\Phi$ denote  Foata&#039;s second fundamental transformation on permutations. For a permutation $\sigma$ in the symmetric group $S_n$, let $\widetilde{\Lambda}_{\sigma}=\{\pi\in S_n\colon\pi\leq_{w} \sigma\}$ be the principal order ideal generated by $\sigma$  in the weak order $\leq_{w}$. Bj ö rner and Wachs have shown that $\widetilde{\Lambda}_{\sigma}$ is invariant under $\Phi$ if and only if $\sigma$ is a 132-avoiding permutation. In this paper, we consider the invariance property of  $\Phi$ on the principal order ideals ${\Lambda}_{\sigma}=\{\pi\in S_n\colon \pi\leq \sigma\}$ with respect to the Bruhat order $\leq$.  We obtain a characterization  of permutations $\sigma$ such that ${\Lambda}_{\sigma}$ are invariant under $\Phi$. We also consider the invariant principal order  ideals with respect to the Bruhat order  under Han&#039;s bijection $H$. We find  that ${\Lambda}_{\sigma}$ is invariant under the bijection $H$ if and only if it is invariant under the transformation $\Phi$."/>
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	<meta name="DC.Identifier" content="v19i4p3"/>
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						<meta name="DC.Subject" xml:lang="en" content="Foata’s second fundamental transformation"/>
								<meta name="DC.Subject" xml:lang="en" content="Han’s bijection"/>
								<meta name="DC.Subject" xml:lang="en" content="Bruhat order"/>
								<meta name="DC.Subject" xml:lang="en" content="principal order ideal"/>
				<meta name="DC.Title" content="Invariant Principal Order Ideals under Foata’s Transformation"/>
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	<meta name="citation_journal_title" content="The Electronic Journal of Combinatorics"/>
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	<meta name="citation_authors" content="Li, Teresa X.S.; Miao, Melissa Y.F."/>
	<meta name="citation_title" content="Invariant Principal Order Ideals under Foata’s Transformation"/>

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						<meta name="citation_keywords" xml:lang="en" content="Foata’s second fundamental transformation"/>
								<meta name="citation_keywords" xml:lang="en" content="Han’s bijection"/>
								<meta name="citation_keywords" xml:lang="en" content="Bruhat order"/>
								<meta name="citation_keywords" xml:lang="en" content="principal order ideal"/>
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