Transitivity And Connectivity Of Permutations

It was observed for years, in particular in quantum physics, that the number of connected permutations of [0; n] (also called indecomposable permutations), i. e. those φ such that for any i < n there exists j > i with φ(j) < i, equals the number of pointed hypermaps of size n, i. e. the number of transitive pairs (σ, θ) of permutations of a set of cardinality n with a distinguished element.

The paper establishes a natural bijection between the two families. An encoding of maps follows.

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Correspondence to Patrice Ossona de Mendez.

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To Chantal, that we may stay connected beyond the simple line of time.

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Ossona de Mendez, P., Rosenstiehl, P. Transitivity And Connectivity Of Permutations. Combinatorica 24, 487–501 (2004). https://doi.org/10.1007/s00493-004-0029-4

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Mathematics Subject Classification (2000):

  • 05A19
  • 05C30