This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Any transposition (or disjoint combination) is one element to be counted.
When n=2, there is only one, and a(2)=1. When n=3, there are only 3 transpositions, but there are other 6 elements, for instance
f:{1,2,3}->{2,1,1} gives fof:{1,2,3}->{1,2,2} and fofof=f (cycle 2),
(the others are similar), thus giving a(3)=9.
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i-1), j=0..n/i)))
end:
A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):
a:= n-> A(n, 2) -A(n, 1):
seq(a(n), n=0..25); # Alois P. Heinz, Aug 19 2014
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[(i - 1)!^j*multinomial[ n, Join[{n - i*j}, Table[i, j]]]/j!*b[n - i*j, i - 1], {j, 0, n/i}]]];
A[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, Min[j, k]], {j, 0, n}];
a[0] = 0; a[n_] := A[n, 2] - A[n, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)
Any period 3 permutation (or disjoint combinations) is one element to be counted.
For n=3, where there are only 2 cases: f1:{1,2,3}->{2,3,1} and f2:{1,2,3}->{3,1,2} but for n>3 there are other elements (non-permutations) to be counted (for instance, with n=5, we count with f:{1,2,3,4,5}->{2,4,5,3,4}).
b:= proc(n, m) option remember; `if`(m>3, 0, `if`(n=0, x^m, add(
(j-1)!*b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n)))
end:
a:= n-> coeff(add(b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n), x, 3):
seq(a(n), n=0..25); # Alois P. Heinz, Aug 14 2017
b[n_, m_] := b[n, m] = If[m>3, 0, If[n == 0, x^m, Sum[(j - 1)! b[n - j, LCM[m, j]] Binomial[n - 1, j - 1], {j, 1, n}]]];
a[n_] := If[n==0, 0, Coefficient[Sum[b[j, 1] n^(n-j) Binomial[n-1, j-1], {j, 0, n}], x, 3]];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)
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