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

0,9

Comments

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

Links

Table of n, a(n) for n=0..65.

Example

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, ...

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 *)

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).

Sequence in context: A010328 A332847 A245501 * A193512 A366524 A366523

Adjacent sequences: A247023 A247024 A247025 * A247027 A247028 A247029

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 09 2014

Status

approved