A068330 - OEIS

%I #13 Feb 11 2024 12:45:07

%S 1,1,1,2,4,6,11,20,36,65,118,215,389,727,1366,2565,4849,9123,17168,

%T 32629,62121,118353,226603,434512,833776,1605642,3101121,5993545,

%U 11593548,22443167,43459975,84209877,163359383

%N 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 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 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a068/A068330.java">Java program</a> (github)

%e 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 (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 Cf. A000048, A056303.

%K nonn,nice,more

%O 1,4

%A Simon P. Norton, Feb 27 2002

%E a(21)-a(33) from _Sean A. Irvine_, Feb 10 2024