Permutations containing and avoiding certain patterns

Abstract

Let T_k^m={σ\in S_k | σ_1=m}. We prove that the number of permutations which avoid all patterns in T_k^m equals (k-2)!(k-1)^{n+1-k} for k <= n. We then prove that for any τin T_k^1 (or any τin T_k^k), the number of permutations which avoid all patterns in T_k^1 (or in T_k^k) except for τand contain τexactly once equals (n+1-k)(k-1)^{n-k} for k <= n. Finally, for any τin T_k^m, 2 <= m <= k-1, this number equals (k-1)^{n-k} for k <= n. These results generalize recent results due to Robertson concerning permutations avoiding 123-pattern and containing 132-pattern exactly once.