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Number of cycles in the graph of 312-avoiding permutations

Ehrenborg, Richard and Kitaev, Sergey and Steingrimsson, Einar (2015) Number of cycles in the graph of 312-avoiding permutations. Journal of Combinatorial Theory Series A, 129. pp. 1-18. ISSN 0097-3165

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Abstract

The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. That is, for every permutation π=π1π2...πn+1 there is a directed edge from the standardization of π1π2...πn to the standardization of π2π3...πn+1. We give a formula for the number of cycles of length d in the subgraph of overlapping 312-avoiding permutations. Using this we also give a refinement of the enumeration of 312-avoiding affine permutations and point out some open problems on this graph, which so far has been little studied.