On Stanley-Wilf limit of the pattern 1324
Abstract
We use the Mansour-Vainshtein theory of kernel shapes to decompose the set \(S_n(1324)\) of 1324-avoiding permutations of length \(n\) into small pieces that are governed by some kernel shape \(\lambda\). An enumeration of the set \(K_{m,c}\) of all kernel shapes with length \(m\) and capacity \(c\) allows to express the generating function for the number of 1324-avoiding permutations of length \(n\) in terms of the \(P_m(C(x)) = \sum_c \lvert K_{m,c+1}\rvert C^c(x)\) where \(P_m(x)\) is a polynomial and \(C(x)\) is the generating function for the Catalan numbers. This allows us to write down a systematic procedure for finding a lower bound for approximating the Stanley-Wilf limit of the pattern 1324. We use an implementation of this method in the OpenCL framework to compute such a bound explicitly.