Surprising symmetries in 132-avoiding permutations

Details Surprising symmetries in 132-avoiding permutations Surprising symmetries in 132-avoiding permutations

2012-02-09

arxiv.org/abs/1202.2023v1

Mathematics

Combinatorics

11 pages, 5 figures

Scientific paper

We prove that the total number $S_{n,132}(q)$ of copies of the pattern $q$ in all 132-avoiding permutations of length $n$ is the same for $q=231$, $q=312$, or $q=213$. We provide a combinatorial proof for this unexpected threefold symmetry. We then significantly generalize this result to show an exponential number of different pairs of patterns $q$ and $q'$ of length $k$ for which $S_{n,132}(q)=S_{n,132}(q')$ and the equality is non-trivial.

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