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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?

%I A068330 #17 May 29 2025 08:01:23
%S A068330 1,1,1,2,4,6,11,20,36,65,118,215,389,727,1366,2565,4849,9123,17168,
%T A068330 32629,62121,118353,226603,434512,833776,1605642,3101121,5993545,
%U A068330 11593548,22443167,43459975,84209877,163359383,317230531,616506533,1199200964,2334860706,4549377408,8870694723
%N 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?
%C A068330 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.
%H A068330 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a068/A068330.java">Java program</a> (github)
%e A068330 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).
%o A068330 (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;
%Y A068330 Cf. A000048, A056303.
%K A068330 nonn,nice
%O A068330 1,4
%A A068330 Simon P. Norton, Feb 27 2002
%E A068330 a(21)-a(33) from _Sean A. Irvine_, Feb 10 2024
%E A068330 a(34)-a(39) from _Bert Dobbelaere_, May 29 2025