Dyck Paths, Standard Young Tableaux, and Pattern Avoiding Permutations
Abstract
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.
Cite
@article{arxiv.0912.4747,
title = {Dyck Paths, Standard Young Tableaux, and Pattern Avoiding Permutations},
author = {Hilmar Gudmundsson},
journal= {arXiv preprint arXiv:0912.4747},
year = {2009}
}
Comments
15 pages, 4 figures. Submitted to a special edition of Pure Mathematics and Applications in 2009. Research supported by grant no. 060005013 from the Icelandic Research Fund