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A244524 Number of pairs (f,g) of commuting maps {0,..,n-1}->{0,..,n-1} with 0 <= f(k), g(k) <= k.

%I A244524 #21 Nov 10 2024 02:24:11
%S A244524 1,1,4,26,236,2780,40642,715836,14873174,358866952,9934283924,
%T A244524 312461402424
%N A244524 Number of pairs (f,g) of commuting maps {0,..,n-1}->{0,..,n-1} with 0 <= f(k), g(k) <= k.
%e A244524 The a(3) = 26 pairs of such maps are (dots for zeros in the maps):
%e A244524 01:  [ . . . ]  [ . . . ]
%e A244524 02:  [ . . . ]  [ . . 1 ]
%e A244524 03:  [ . . . ]  [ . . 2 ]
%e A244524 04:  [ . . . ]  [ . 1 . ]
%e A244524 05:  [ . . . ]  [ . 1 1 ]
%e A244524 06:  [ . . . ]  [ . 1 2 ]
%e A244524 07:  [ . . 1 ]  [ . . . ]
%e A244524 08:  [ . . 1 ]  [ . . 1 ]
%e A244524 09:  [ . . 1 ]  [ . 1 2 ]
%e A244524 10:  [ . . 2 ]  [ . . . ]
%e A244524 11:  [ . . 2 ]  [ . . 2 ]
%e A244524 12:  [ . . 2 ]  [ . 1 . ]
%e A244524 13:  [ . . 2 ]  [ . 1 2 ]
%e A244524 14:  [ . 1 . ]  [ . . . ]
%e A244524 15:  [ . 1 . ]  [ . . 2 ]
%e A244524 16:  [ . 1 . ]  [ . 1 . ]
%e A244524 17:  [ . 1 . ]  [ . 1 2 ]
%e A244524 18:  [ . 1 1 ]  [ . . . ]
%e A244524 19:  [ . 1 1 ]  [ . 1 1 ]
%e A244524 20:  [ . 1 1 ]  [ . 1 2 ]
%e A244524 21:  [ . 1 2 ]  [ . . . ]
%e A244524 22:  [ . 1 2 ]  [ . . 1 ]
%e A244524 23:  [ . 1 2 ]  [ . . 2 ]
%e A244524 24:  [ . 1 2 ]  [ . 1 . ]
%e A244524 25:  [ . 1 2 ]  [ . 1 1 ]
%e A244524 26:  [ . 1 2 ]  [ . 1 2 ]
%p A244524 s:= proc(n) option remember; `if`(n=0, [[]],
%p A244524        map(x-> seq([x[], i], i=1..n), s(n-1)))
%p A244524     end:
%p A244524 a:= n-> (l-> add(add(`if`([seq(evalb(f[g[i]]=g[f[i]])
%p A244524     , i=1..n)]=[true$n], 1, 0), g=l), f=l))(s(n)):
%p A244524 seq(a(n), n=0..6);  # _Alois P. Heinz_, Jul 30 2014
%Y A244524 Cf. A181162 (commuting maps {1,..,n}->{1,..,n} without restrictions).
%Y A244524 Cf. A053529 (commuting permutations).
%K A244524 nonn,hard,more
%O A244524 0,3
%A A244524 _Joerg Arndt_, Jul 25 2014
%E A244524 All terms corrected (error pointed out by _Alois P. Heinz_), _Joerg Arndt_, Jul 30 2014
%E A244524 a(10)-a(11) from _Alois P. Heinz_, Jul 30 2014