Shape-Wilf-Ordering on Permutations of Length 3

The research on pattern-avoidance has yielded so far limited knowledge on Wilfordering 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 complete ordering of Sk for k = 4: |Sn(1342)| < |Sn(1234)| < |Sn(1324)| for n ≥ 7, along with some other sporadic examples in Wilf-ordering. 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 Wilfequivalent, the current paper distinguishes between and orders these permutations by employing all Young diagrams. This opens up the question of whether shapeWilf-ordering of permutations, or some generalization of it, is not the “true” way of approaching pattern-avoidance ordering.