Journal of Combinatorics

Volume 6 (2015)

Number 1–2

Some Wilf-equivalences for vincular patterns

Pages: 19 – 45



Andrew M. Baxter (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Mark Shattuck (Department of Mathematics, University of Tennessee, Knoxville, Tenn., U.S.A.)


We prove several Wilf-equivalences for vincular patterns of length $4$, some of which generalize to infinite families of vincular patterns. We also present functional equations for the generating functions for the number of permutations of length $n$ avoiding a single pattern for the patterns 124-3, 134-2, 231-4, 241-3, 132-4, and 142-3. This nearly completes the Wilf-classification of vincular patterns of length 4. As a corollary, these results imply Wilf-equivalences for certain barred patterns of length 5 with a single bar.


permutations, pattern avoidance, vincular pattern, Wilf-equivalence

Published 20 March 2015