New equivalences for pattern avoiding involutions

Abstract

We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard’s conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 5"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 6"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>6</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 7"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>7</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for involutions.