Shape-Wilf-ordering of permutations of length 3

Abstract

The research on pattern-avoidance has yielded so far limited knowledge on Wilf-ordering of permutations. The Stanley-Wilf limits sqrt[n](|S_n(tau)|) and further works suggest asymptotic ordering of layered versus monotone patterns. Yet, Bona has provided the only known up to now result of its type on ordering of permutations: |S_n(1342)|<|S_n(1234)|<|S_n(1324)| for n>6. We give a different proof of this result by ordering S_3 up to the stronger shape-Wilf-order: |S_Y(213)|<=|S_Y(123)|<=|S_Y(312)| for any Young diagram Y, derive as a consequence that |S_Y(k+2,k+1,k+3,tau)|<=|S_Y(k+1,k+2,k+3,tau)|<= |S_Y(k+3,k+1,k+2,tau)| for any tau in S_k, and find out when equalities are obtained. (In particular, for specific Y's we find out that |S_Y(123)|=|S_Y(312)| coincide with every other Fibonacci term.) This strengthens and generalizes Bona'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.