Pattern avoidance by even permutations (Q665763)

Summary: We study questions of even-Wilf-equivalence, the analogue of Wilf-equivalence when attention is restricted to pattern avoidance by permutations in the alternating group. Although some Wilf-equivalence results break when considering even-Wilf-equivalence analogues, we prove that other Wilf-equivalence results continue to hold in the even-Wilf-equivalence setting. In particular, we prove that \(t(t - 1) \cdot \cdot \cdot 321\) and \((t - 1)(t - 2) \cdot \cdot \cdot \)21t are even-shape-Wilf-equivalent for odd \(t\), paralleling a result (which held for all \(t\)) of \textit{J. Backelin}, \textit{J. West}, and \textit{G. Xin} [Adv. Appl. Math. 38, No. 2, 133--148 (2007; Zbl 1127.05002)] for shape-Wilf-equivalence. This allows us to classify the symmetric group \(\mathcal S_4\), and to partially classify \(\mathcal S_5\) and \(\mathcal S_6\), according to even-Wilf-equivalence. As with transition to involution-Wilf-equivalence, some-but not all-of the classical Wilf-equivalence results are preserved when we make the transition to even-Wilf-equivalence.