A239773 Number of pairs of functions f, g from a size n set into itself satisfying f(g(f(x))) = f(f(f(x))).

1, 1, 10, 231, 9880, 644845, 58790736, 7077540295, 1081801600384, 203836779804537, 46268684631596800

Offset: 0

Author

Chad Brewbaker, Mar 26 2014

Keywords

• nonn
• more

Crossrefs

Cf. A181162.

Programs

Maple

s:= proc(n, i) option remember; `if`(i=0, [[]],
       map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
    end:
a:= proc(n) (l-> add(add(`if`([true$n]=[seq(evalb(
       f[g[f[i]]]=f[f[f[i]]]), i=1..n)], 1, 0), g=l), f=l))(s(n$2))
    end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014

Mathematica

a[n_] := a[n] = If[n == 0, 1, Module[{f, g, T}, T = Tuples[Range[n], n]; Table[f = T[[j, #]] &; g = T[[k, #]] &; Table[True, {n}] == Table[f[g[f[i]]] == f[f[f[i]]], {i, n}], {j, n^n}, {k, n^n}] // Flatten // Count[#, True] &]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 5}] (* Jean-François Alcover, Sep 24 2022 *)

Extensions

a(6)-a(7) from Giovanni Resta, Mar 28 2014

a(8)-a(10) from Max Alekseyev, Dec 19 2024