cp's OEIS Frontend

A247026 Number A(n,k) of endofunctions on [n] that are the k-th power of an endofunction; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A247026 #17 May 05 2019 08:58:23
%S A247026 1,1,1,1,1,1,1,1,4,1,1,1,3,27,1,1,1,4,12,256,1,1,1,3,19,100,3125,1,1,
%T A247026 1,4,12,116,1075,46656,1,1,1,3,21,73,985,13356,823543,1,1,1,4,10,148,
%U A247026 580,11026,197764,16777216,1,1,1,3,21,44,1281,5721,145621,3403576,387420489,1
%N A247026 Number A(n,k) of endofunctions on [n] that are the k-th power of an endofunction; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A247026 Number of endofunctions f on [n] such that an endofunction g on [n] exists with f=g^k.
%e A247026 A(3,2) = 12: (1,1,1), (1,1,3), (1,2,1), (1,2,2), (1,2,3), (1,3,3), (2,2,2), (2,2,3), (2,3,1), (3,1,2), (3,2,3), (3,3,3).
%e A247026 A(3,6) = 10: (1,1,1), (1,1,3), (1,2,1), (1,2,2), (1,2,3), (1,3,3), (2,2,2), (2,2,3), (3,2,3), (3,3,3).
%e A247026 A(4,4) = 73: (1,1,1,1), (1,1,1,4), (1,1,3,1), (1,1,3,3), ..., (4,4,1,3), (4,4,2,3), (4,4,3,4), (4,4,4,4).
%e A247026 Square array A(n,k) begins:
%e A247026   1,      1,      1,      1,     1,      1,     1,      1, ...
%e A247026   1,      1,      1,      1,     1,      1,     1,      1, ...
%e A247026   1,      4,      3,      4,     3,      4,     3,      4, ...
%e A247026   1,     27,     12,     19,    12,     21,    10,     21, ...
%e A247026   1,    256,    100,    116,    73,    148,    44,    148, ...
%e A247026   1,   3125,   1075,    985,   580,   1281,   295,   1305, ...
%e A247026   1,  46656,  13356,  11026,  5721,  12942,  3136,  13806, ...
%e A247026   1, 823543, 197764, 145621, 69244, 150955, 42784, 169681, ...
%t A247026 (* This program is not suitable to compute a large number of terms. *)
%t A247026 nmax = 8;
%t A247026 f[a_][b_] /; Length[a]==Length[b] := Table[b[[a[[i]]]], {i, 1, Length[a]}];
%t A247026 A[n_, k_] := Nest[f[#], Range[n], k]& /@ Tuples[Range[n], {n}] // Union // Length;
%t A247026 Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, May 05 2019 *)
%Y A247026 Columns k=0-10 give: A000012, A000312, A102687, A163859, A163860, A163861, A247053, A247054, A247055, A247056, A247057.
%Y A247026 Rows n=0+1, 2-7 give: A000012, A103947, A103948, A103949, A102709, A103950, A247058.
%Y A247026 Main diagonal gives A247059.
%Y A247026 Cf. A247005 (the same for permutations).
%K A247026 nonn,tabl
%O A247026 0,9
%A A247026 _Alois P. Heinz_, Sep 09 2014