Permutations Containing and Avoiding 123 and 132 Patterns

We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern equals (n-2)2^(n-3). We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is (n-3)(n-4)2^(n-5).Comment: 5 page

New Lower Bounds for Some Multicolored Ramsey Numbers

We use finite fields and extend a result of Fan Chung to give eight new, nontrivial, lower bounds.Comment: 6 page

Permutations Restricted by Two Distinct Patterns of Length Three

Define $S_n(R;T)$ to be the number of permutations on $n$ letters which avoid all patterns in the set $R$ and contain each pattern in the multiset $T$ exactly once. In this paper we enumerate $S_n(\{\alpha\};\{\beta\})$ and $S_n(\emptyset;\{\alpha,\beta\})$ for all $\alpha \neq \beta \in S_3$. The results for $S_n(\{\alpha\};\{\beta\})$ follow from two papers by Mansour and Vainshtein.Comment: 15 pages, some relevant reference brought to my attention (see section 4

Difference Ramsey Numbers and Issai Numbers

We present a recursive algorithm for finding good lower bounds for the classical Ramsey numbers. Using notions from this algorithm we then give some results for generalized Schur numbers, which we call Issai numbers.Comment: 10 page