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            <title xml:lang="en">Equivalence Relations of Permutations Generated by Constrained Transpositions</title>
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                  <title>22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)</title>
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                <editor>Billey</editor>
                <editor>Sara and Reiner</editor>
                <editor>Victor</editor>
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                <term xml:lang="en">pattern-avoiding permutations</term>
                <term xml:lang="en">connected permutations</term>
                <term xml:lang="en">layered permutations</term>
                <term xml:lang="en">Catalan numbers</term>
                <term xml:lang="en">integer sequences</term>
                <term xml:lang="en">Knuth relations</term>
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              <p>We consider a large family of equivalence relations on permutations in $S_n$ that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, conditional upon the presence of a third element of suitable value and location. For some relations of this type, we compute the number of equivalence classes, determine how many $n$-permutations are equivalent to the identity permutation, or characterise this equivalence class. Although our results include familiar integer sequences (e.g., Catalan, Fibonacci, and Tribonacci numbers) and special classes of permutations (layered, connected, and $123$-avoiding), some of the sequences that arise appear to be new.</p>
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              <p>Nous considérons une famille de relations d’équivalence sur l'ensemble $S_n$ des permutations, qui généralisent les relations de Knuth liées à la correspondance Robinson-Schensted. Dans notre contexte général, deux permutations sont considérées comme équivalentes si l'une peut être obtenue de l'autre auprès d'une séquence de remplacements d'un motif par un autre selon des règles précisées. Désormais, nous ne considérons dans l’œuvre actuelle que les motifs qui correspondent à la transposition de deux éléments, conditionné sur la présence d'un élément de valeur et de position approprié. Pour plusieurs exemples de ce problème, nous énumérons les classes d'équivalence, nous déterminons combien de permutations sur $n$ éléments sont équivalentes à l'identité, ou nous précisons la forme des éléments dans cette dernière classe. Bien que nos résultats retrouvent des séquences des entiers très bien connues (nombres de Catalan, de Fibonacci, de Tribonacci...) ainsi que des classes de permutations déjà étudiées (en couches, connexes, sans motif $123$), nous trouvons également des séquences qui paraissent être nouvelles.</p>
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