Shape - Wilf - Ordering on Permutations of Length

The research on pattern-avoidance has yielded so far limited knowledge on Wilf-ordering of permutations. The Stanley-Wilf limits limn→∞ n √ |Sn(τ )| and further works suggest asymptotic ordering of layered versus monotone patterns. Yet, Bóna has provided essentially the only known up to now result of its type on ordering of permutations: |Sn(1342)| < |Sn(1234)| < |Sn(1324)| for n ≥ 7. We give a different proof of this result by ordering S3 up to the stronger shape-Wilf-order: |SY (213)| ≤ |SY (123)| ≤ |SY (312)| for any Young diagram Y , derive as a consequence that |SY (k + 2, k + 1, k + 3, τ )| ≤ |SY (k+1, k+2, k+3, τ )| ≤ |SY (k+3, k+1, k+2, τ )| for any τ ∈ Sk, and find out when equalities are obtained. (In particular, for specific Y ’s we find out that |SY (123)| = |SY (312)| coincide with every other Fibonacci term.) This strengthens and generalizes Bóna’s result to arbitrary length permutations. While all length-3 permutations have been shown in numerous ways to be Wilf-equivalent, the current paper distinguishes between and orders these permutations by employing all Young diagrams. This opens up the question of whether shape-Wilf-ordering of permutations, or some generalization of it, is not the “true” way of approaching pattern-avoidance ordering.