On permutations which avoid the patterns 1324 and 2143 - Steven Butler

We consider permutations which avoid the patterns 1324 and 2143, where by 2143 we mean it avoids the pattern 2143 with Bruhat restriction {2<->3}. For each permutation p we can construct a linear map L_p and a graph G_p. Our main result will be to show that the following are equivalent: p avoids 1324 and 2143; L_p is onto; G_p is a forest. This proves a conjecture of Woo and Yong.

Time allowing we will explore some interesting properties of G_p, and show that there are 2^n-1-1 permutations in S_n whose graph is a path.