A239780
Number of pairs of functions f, g on a size n set into itself satisfying f(f(g(x))) = f(g(g(x))).
Original entry on oeis.org
1, 1, 12, 279, 11128, 669605, 55937376, 6146973259
Offset: 0
-
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[f[g[i]]]=f[g[g[i]]]), i=1..n)], 1, 0), g=l), f=l))(s(n$2))
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 17 2014
A239782
Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(f(x))) = f(f(g(x))).
Original entry on oeis.org
1, 1, 14, 321, 12256, 682985, 52551216, 5333977789
Offset: 0
A239783
Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(x)) = f(g(f(x))).
Original entry on oeis.org
1, 1, 14, 339, 12496, 639005, 42829536, 3624387739
Offset: 0
A239784
Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(g(x))) = f(g(f(x))).
Original entry on oeis.org
1, 1, 14, 351, 14776, 902605, 75313056, 8194699387
Offset: 0
A244524
Number of pairs (f,g) of commuting maps {0,..,n-1}->{0,..,n-1} with 0 <= f(k), g(k) <= k.
Original entry on oeis.org
1, 1, 4, 26, 236, 2780, 40642, 715836, 14873174, 358866952, 9934283924, 312461402424
Offset: 0
The a(3) = 26 pairs of such maps are (dots for zeros in the maps):
01: [ . . . ] [ . . . ]
02: [ . . . ] [ . . 1 ]
03: [ . . . ] [ . . 2 ]
04: [ . . . ] [ . 1 . ]
05: [ . . . ] [ . 1 1 ]
06: [ . . . ] [ . 1 2 ]
07: [ . . 1 ] [ . . . ]
08: [ . . 1 ] [ . . 1 ]
09: [ . . 1 ] [ . 1 2 ]
10: [ . . 2 ] [ . . . ]
11: [ . . 2 ] [ . . 2 ]
12: [ . . 2 ] [ . 1 . ]
13: [ . . 2 ] [ . 1 2 ]
14: [ . 1 . ] [ . . . ]
15: [ . 1 . ] [ . . 2 ]
16: [ . 1 . ] [ . 1 . ]
17: [ . 1 . ] [ . 1 2 ]
18: [ . 1 1 ] [ . . . ]
19: [ . 1 1 ] [ . 1 1 ]
20: [ . 1 1 ] [ . 1 2 ]
21: [ . 1 2 ] [ . . . ]
22: [ . 1 2 ] [ . . 1 ]
23: [ . 1 2 ] [ . . 2 ]
24: [ . 1 2 ] [ . 1 . ]
25: [ . 1 2 ] [ . 1 1 ]
26: [ . 1 2 ] [ . 1 2 ]
Cf.
A181162 (commuting maps {1,..,n}->{1,..,n} without restrictions).
Cf.
A053529 (commuting permutations).
-
s:= proc(n) option remember; `if`(n=0, [[]],
map(x-> seq([x[], i], i=1..n), s(n-1)))
end:
a:= n-> (l-> add(add(`if`([seq(evalb(f[g[i]]=g[f[i]])
, i=1..n)]=[true$n], 1, 0), g=l), f=l))(s(n)):
seq(a(n), n=0..6); # Alois P. Heinz, Jul 30 2014