A254569
The number of unordered pairs (f,g) of functions from {1..n} to itself such that fg=gf (i.e., f(g(i))=g(f(i)) for all i).
Original entry on oeis.org
1, 7, 84, 1540, 37345, 1145376, 42402871, 1849021504, 93217426857, 5363120671120, 348669188664511, 25418305492373520, 2064813985107357445, 185896884170249831320, 18459391640792004885375, 2012607682674617326564096, 239898601216105901349115537, 31132586664410794285693925664, 4380971763246528510240944123071, 665896706682993760478978112600400
Offset: 1
The a(2) = 7 pairs of maps [2] -> [2] are:
01: [ 1 1 ] [ 1 1 ]
02: [ 1 1 ] [ 1 2 ]
03: [ 1 2 ] [ 1 2 ]
04: [ 1 2 ] [ 2 1 ]
05: [ 1 2 ] [ 2 2 ]
06: [ 2 1 ] [ 2 1 ]
07: [ 2 2 ] [ 2 2 ]
Cf.
A181162 (ordered pairs),
A254570 (unordered pairs, f and g distinct).
A277337
Number of pairs of functions (f,g) from a set of n elements into itself that are generalized reflexive inverses of each other.
Original entry on oeis.org
1, 1, 6, 87, 2056, 71145, 3355956, 203899087, 15451934016, 1419181414929, 154796303577700, 19713331210664751, 2891162097251141616, 482733064744447450297, 90871916094948544512516, 19125402877558442317308975, 4467829768503489097383022336, 1151133088512781095709101702177, 325279313240363190497696752254276
Offset: 0
For n=2 the a(2)=6 solutions are
1: [1,1] [1,1]
2: [1,1] [2,2]
3: [2,2] [1,1]
4: [2,2] [2,2]
5: [1,2] [1,2]
6: [2,1] [2,1]
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Flatten[{1, Table[Sum[n!*Binomial[n, k]*k^(2*(n-k))/(n-k)!, {k, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 21 2016 *)
-
a(n) = sum(k = 1, n, n! / (n - k)! * binomial(n, k) * k^(2 * (n - k) ) ); \\ Joerg Arndt, Oct 10 2016
A239762
Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(g(x))) = g(f(x)).
Original entry on oeis.org
1, 1, 10, 159, 3568, 106545, 4062336, 192754009
Offset: 0
A239763
Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(x)) = f(g(x)).
Original entry on oeis.org
1, 1, 10, 171, 4600, 168285, 7988736, 472245991
Offset: 0
A239764
Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(f(x))) = f(g(x)).
Original entry on oeis.org
1, 1, 10, 183, 5128, 203085, 10709136, 722183371
Offset: 0
A239765
Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(g(x))) = g(g(g(x))).
Original entry on oeis.org
1, 1, 10, 189, 6304, 315065, 21848976, 1992930037
Offset: 0
A239766
Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(x)) = f(g(g(x))).
Original entry on oeis.org
1, 1, 10, 195, 5992, 260085, 14922576, 1087100371
Offset: 0
A239770
Number of pairs of functions f, g from a size n set into itself satisfying f(g(f(x))) = g(f(f(x))).
Original entry on oeis.org
1, 1, 10, 213, 7720, 420865, 31879296, 3175850965
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) local l; l:= s(n$2);
add(add(`if`([seq(evalb(f[g[f[i]]]=g[f[f[i]]]),
i=1..n)]=[true$n], 1, 0), g=l), f=l)
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
A239774
Number of pairs of functions f, g from a size n set into itself satisfying f(f(g(x))) = f(f(f(x))).
Original entry on oeis.org
1, 1, 10, 285, 14176, 1034145, 105764256, 14367333421
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) local l; l:= s(n$2);
add(add(`if`([seq(evalb(f[f[g[i]]]=f[f[f[i]]]),
i=1..n)]=[true$n], 1, 0), g=l), f=l)
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
A239776
Number of pairs of functions f, g from a size n set into itself satisfying f(f(g(x))) = g(g(f(x))).
Original entry on oeis.org
1, 1, 12, 189, 5200, 208945, 11517936, 828676933
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]]]=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
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