Pattern avoidance by even permutations

Andrew Baxter ; Aaron D. Jaggard

The electronic journal of combinatorics, The Zeilberger Festschrift volume, Tome 18 (2011) no. 2

Résumé

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 $t(t-1)\cdots 321$ and $(t-1)(t-2)\cdots 21t$ are even-shape-Wilf-equivalent for odd $t$, paralleling a result (which held for all $t$) of Backelin, West, and Xin 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.

Détail

DOI : 10.37236/2024

Classification : 05A05, 05A15, 05A19, 05B20
Mots-clés : permutation pattern, Wilf-equivalence, even-Wilf-equivalence, alternating group, even permutation