Geodesic
Permutations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 64 (2009) no. 4, pp. 583-624

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Decompositions into cycles for random permutations of a large number of elements are very different (in their statistics) from the same decompositions for algebraic permutations (defined by linear or projective transformations of finite sets). This paper presents tables giving both these and other statistics, as well as a comparison of them with the statistics of involutions or permutations with all their cycles of even length. The inclusions of a point in cycles of various lengths turn out to be equiprobable events for random permutations. The number of permutations of $2N$ elements with all cycles of even length turns out to be the square of an integer (namely, of $(2N-1)!!$). The number of cycles of projective permutations (over a field with an odd prime number of elements) is always even. These and other empirically discovered theorems are proved in the paper. Bibliography: 6 titles.
Keywords: Young diagrams, symmetric group, modular group, projective geometry, statistics, randomness.
Mots-clés : cycles, involutions 
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     author = {V. I. Arnold},
     title = {Permutations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     number = {4},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2009_64_4_a0/}
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V. I. Arnold. Permutations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 64 (2009) no. 4, pp. 583-624. http://geodesic.mathdoc.fr/item/RM_2009_64_4_a0/