Generating trees for permutations avoiding generalized patterns

We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-M\'elou. We obtain refinements of known enumerative results and find new ones.Comment: 17 pages, to appear in Ann. Com

Recurrence relations for patterns of type (2,1)(2,1) in flattened permutations

We consider the problem of counting the occurrences of patterns of the form $xy-z$ within flattened permutations of a given length. Using symmetric functions, we find recurrence relations satisfied by the distributions on $\mathcal{S}_n$ for the patterns 12-3, 21-3, 23-1 and 32-1, and develop a unified approach to obtain explicit formulas. By these recurrences, we are able to determine simple closed form expressions for the number of permutations that, when flattened, avoid one of these patterns as well as expressions for the average number of occurrences. In particular, we find that the average number of 23-1 patterns and the average number of 32-1 patterns in $\text{Flatten}(\pi)$, taken over all permutations $\pi$ of the same length, are equal, as are the number of permutations avoiding either of these patterns. We also find that the average number of 21-3 patterns in $\text{Flatten}(\pi)$ over all $\pi$ is the same as it is for 31-2 patterns.Comment: 19 pages. Final version will be published in Journal of Difference Equations and Application