A239769 Number of pairs of functions (f, g) from a size n set into itself satisfying f(g(f(x))) = g(g(f(x))).

1, 1, 10, 195, 7000, 397445, 32540976, 3612881587, 520731462784

Offset: 0

Author

Chad Brewbaker, Mar 26 2014

Keywords

• nonn
• more

Crossrefs

Cf. A181162, A239773.

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]]]=g[g[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]]] == g[g[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 22 2022 *)

Extensions

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

a(8) from Max Alekseyev, Dec 20 2024