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A239841
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Ordered pairs of permutation functions on n elements where f(g(g(x))) = g(g(f(x))).
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6
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1, 1, 4, 30, 312, 3720, 64080, 1305360, 33949440, 1019692800, 36360576000, 1487539468800, 69633899596800, 3649476307276800, 213929162589542400, 13848506938506240000, 986705192227442688000, 76724136092268048384000, 6491471142159880740864000
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OFFSET
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0,3
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COMMENTS
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Suppose G is the symmetric group on n letters. For each g in G, the set of f satisfying fgg = ggf is just the centralizer Z_gg(G). However |Z_gg(G)| is clearly constant on conjugacy classes of G. By the orbit-stabilizer theorem the size of the conjugacy class containing g is |G| / |Z_g(G)|. Since |G| = n! and Z_g(G) is a subgroup of Z_gg(G) we see that a(n) equals n! multiplied by the sum of indices |Z_gg(G) : Z_g(G)| where the sum is over representatives of the conjugacy classes of G. Since the conjugacy classes of G correspond to partitions of n (A000041), this makes it relatively easy to find terms.
a(n) appears to equal n! * A082733(n).
(End)
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REFERENCES
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John F. Humphreys, A Course In Group Theory, Oxford Science Publications, 1996, chapter 10.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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