<p>In 2019, Bóna and Smith introduced the notion of <i>strong pattern avoidance</i> for which a permutation and its square both avoid a given pattern. In this paper, we enumerate the set of permutations <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2024_2881_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> which not only strongly avoid the pattern 312 or 231 but also avoid the pattern <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2024_2881_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>, for all <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2024_2881_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau \in S_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>∈</mo> <msub> <mi>S</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and some <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2024_2881_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau \in S_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>∈</mo> <msub> <mi>S</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. One of these results gives a positive answer to a conjecture of Archer and Geary.</p>

On the Permutations that Strongly Avoid the Pattern 312 or 231

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发表日期：

2024年12月31日

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Junyao Pan,
Pengfei Guo

摘要

In 2019, Bóna and Smith introduced the notion of strong pattern avoidance for which a permutation and its square both avoid a given pattern. In this paper, we enumerate the set of permutations $\pi $ $π$ which not only strongly avoid the pattern 312 or 231 but also avoid the pattern $\tau $ $τ$ , for all $\tau \in S_3$ $τ \in S_{3}$ and some $\tau \in S_4$ $τ \in S_{4}$ . One of these results gives a positive answer to a conjecture of Archer and Geary.