Bóna, Miklós Surprising symmetries in objects counted by Catalan numbers. (English) Zbl 1243.05006 Electron. J. Comb. 19, No. 1, Research Paper P62, 11 p. (2012). 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 \(S_{n,132}(q)=S_{n,132}(q')\). Cited in 4 ReviewsCited in 18 Documents MSC: 05A05 Permutations, words, matrices 05A15 Exact enumeration problems, generating functions Keywords:permutations; patterns; plane trees; bijection PDFBibTeX XMLCite \textit{M. Bóna}, Electron. J. Comb. 19, No. 1, Research Paper P62, 11 p. (2012; Zbl 1243.05006) Full Text: EMIS