Surprising symmetries in objects counted by Catalan numbers (Q426837)
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scientific article; zbMATH DE number 6045686
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Surprising symmetries in objects counted by Catalan numbers |
scientific article; zbMATH DE number 6045686 |
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Surprising symmetries in objects counted by Catalan numbers (English)
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12 June 2012
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Summary: 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 by proving a large family of non-trivial equalities of the type \(S_{n,132}(q)=S_{n,132}(q')\).
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permutations
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patterns
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plane trees
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bijection
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