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A068330
Consider all sublists of [(2,1),(3,2,1),(4,3,2,1),...,(n,...,4,3,2,1)] and multiply these permutations in that order. How many of the products are n-cycles?
1
1, 1, 1, 2, 4, 6, 11, 20, 36, 65, 118, 215, 389, 727, 1366, 2565, 4849, 9123, 17168, 32629, 62121, 118353, 226603, 434512, 833776, 1605642, 3101121, 5993545, 11593548, 22443167, 43459975, 84209877, 163359383, 317230531, 616506533, 1199200964, 2334860706, 4549377408, 8870694723
OFFSET
1,4
COMMENTS
If we take the inverse permutations to the above, or, equivalently, multiply them in the reverse order, we get another description of the sequences A000048 or A056303 with the first term omitted in each case.
LINKS
Sean A. Irvine, Java program (github)
EXAMPLE
a[5] (the output of the program below in which a is the list of the first n terms of the sequence) is 4 because that is the number of products of sublists of [(2,1),(3,2,1),(4,3,2,1),(5,4,3,2,1)] which are 5-cycles, namely (5,4,3,2,1) itself, (3,2,1)*(5,4,3,2,1) = (5,4,3,1,2), (2,1)*(4,3,2,1)*(5,4,3,2,1) = (5,4,2,3,1) and (2,1)*(3,2,1)*(4,3,2,1)*(5,4,3,2,1) = (5,4,2,1,3).
PROG
(GAP) a := []; p := (); perms := [p]; for i in [1..n] do pp := perms*p; pp1 := Filtered(pp, m -> CycleLength(m, [1..i], 1) = i); a[i] := Length(pp1); perms := Union(perms, pp); p := p*(i, i+1); od;
CROSSREFS
Sequence in context: A026385 A254532 A199926 * A017993 A049870 A093970
KEYWORD
nonn,nice
AUTHOR
Simon P. Norton, Feb 27 2002
EXTENSIONS
a(21)-a(33) from Sean A. Irvine, Feb 10 2024
a(34)-a(39) from Bert Dobbelaere, May 29 2025
STATUS
approved