Dyck Paths, Standard Young Tableaux, and Pattern Avoiding Permutations

Details Dyck Paths, Standard Young Tableaux, and Pattern Avoiding Permutations Dyck Paths, Standard Young Tableaux, and Pattern Avoiding Permutations

2009-12-23

arxiv.org/abs/0912.4747v1

Mathematics

Combinatorics

15 pages, 4 figures. Submitted to a special edition of Pure Mathematics and Applications in 2009. Research supported by grant

Scientific paper

We present a generating function and a closed counting formula in two variables that enumerate a family of classes of permutations that avoid or contain an increasing pattern of length three and have a prescribed number of occurrences of another pattern of length three. This gives a refinement of some previously studied statistics, most notably one by Noonan. The formula is also shown to enumerate a family of classes of Dyck paths and Standard Young Tableaux, and a bijection is given between the corresponding classes of these two families of objects. Finally, the results obtained are used to solve an optimization problem for a certain card game.

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