Permutations containing and avoiding certain patterns

Let T-k(m) = {sigma is an element of S-k \ sigma(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 less than or equal to n. We then prove that for any tau is an element of T-k(1) (or any tau is an element of T-k(k)), the number of permutations which avoid all patterns in T-k(1) (or in T-k(k)) except for tau and contain tau exactly once equals (n+1-k)(k-1)(n-k) for k less than or equal to n. Finally, for any tau is an element of T-k(m), 2 less than or equal to m less than or equal to k - 1, this number equals (k - 1)(n-k) for k less than or equal to n. These results generalize recent results due to Robertson concerning permutations avoiding 123-pattern and containing 132-pattern exactly once.