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.

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, 1, 4, 12, 116, 1075, 46656, 1, 1, 1, 3, 21, 73, 985, 13356, 823543, 1, 1, 1, 4, 10, 148, 580, 11026, 197764, 16777216, 1, 1, 1, 3, 21, 44, 1281, 5721, 145621, 3403576, 387420489, 1

Offset: 0

Author

Alois P. Heinz, Sep 09 2014

Keywords

• nonn
• tabl

Comments

Number of endofunctions f on [n] such that an endofunction g on [n] exists with f=g^k.

Examples

			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).
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).
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).
Square array A(n,k) begins:
  1,      1,      1,      1,     1,      1,     1,      1, ...
  1,      1,      1,      1,     1,      1,     1,      1, ...
  1,      4,      3,      4,     3,      4,     3,      4, ...
  1,     27,     12,     19,    12,     21,    10,     21, ...
  1,    256,    100,    116,    73,    148,    44,    148, ...
  1,   3125,   1075,    985,   580,   1281,   295,   1305, ...
  1,  46656,  13356,  11026,  5721,  12942,  3136,  13806, ...
  1, 823543, 197764, 145621, 69244, 150955, 42784, 169681, ...
		

Crossrefs

Columns k=0-10 give: A000012, A000312, A102687, A163859, A163860, A163861, A247053, A247054, A247055, A247056, A247057.

Rows n=0+1, 2-7 give: A000012, A103947, A103948, A103949, A102709, A103950, A247058.

Main diagonal gives A247059.

Cf. A247005 (the same for permutations).

Programs

Mathematica

(* This program is not suitable to compute a large number of terms. *)
nmax = 8;
f[a_][b_] /; Length[a]==Length[b] := Table[b[[a[[i]]]], {i, 1, Length[a]}];
A[n_, k_] := Nest[f[#], Range[n], k]& /@ Tuples[Range[n], {n}] // Union // Length;
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 05 2019 *)