A102709 -id:A102709 - 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.

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

Alois P. Heinz, Sep 09 2014

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

			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, ...

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).

(* 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 *)

A102687 Number of different squares of labeled mappings of a finite set of n elements into itself.

Original entry on oeis.org

1, 1, 3, 12, 100, 1075, 13356, 197764, 3403576, 66159405, 1438338070

Offset: 0

Eric Wegrzynowski (Eric.Wegrzynowski(AT)lifl.fr), Feb 03 2005

Let A be a finite set of cardinal n, F be the set of mappings from A to A and F_2 be the subset of F including all g such that there exists f in F with g = fof (composition of f with itself). Then a(n) = #F_2.

Cf. A102709.

Column k=2 of A247026.

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;
a[n_] := a[n] = A[n, 2];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 7}] (* Jean-François Alcover, May 27 2019 *)

a(7) from Vladeta Jovovic, Feb 05 2005
a(8) and a(9) from Joshua Zucker, May 18 2006
a(0) from Alois P. Heinz, Sep 09 2014
a(10) from Bert Dobbelaere, Jan 24 2019

A103950 a(n) is the number of distinct n-th powers of functions {1, 2, 3, 4, 5, 6} -> {1, 2, 3, 4, 5, 6}.

Original entry on oeis.org

1, 46656, 13356, 11026, 5721, 12942, 3136, 13806, 5601, 9286, 5952, 13806, 1921, 13806, 6816, 8422, 5601, 13806, 3136, 13806, 4737, 9286, 6816, 13806, 1921, 12942, 6816, 9286, 5601, 13806, 2272, 13806, 5601, 9286, 6816, 12942, 1921, 13806, 6816

Offset: 0

David Wasserman, Feb 23 2005

For n > 4, a(n+60) = a(n).

Ray Chandler, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-1, -2, -2, -2, -1, 0, 1, 2, 2, 2, 1, 1).

Cf. A102687, A102709, A103947-A103949.

Row n=6 of A247026.

Join[{1, 46656, 13356, 11026, 5721},LinearRecurrence[{-1, -2, -2, -2, -1, 0, 1, 2, 2, 2, 1, 1},{12942, 3136, 13806, 5601, 9286, 5952, 13806, 1921, 13806, 6816, 8422, 5601},34]] (* Ray Chandler, Sep 08 2015 *)

G.f.: (120*x^16 +1860*x^15 +8520*x^14 +43110*x^13 +48594*x^12 +67518*x^11 +57786*x^10 +26294*x^9 -27373*x^8 -89675*x^7 -122940*x^6 -160740*x^5 -136773*x^4 -117696*x^3 -60014*x^2 -46657*x -1) / ((x -1)*(x +1)*(x^2 -x +1)*(x^2 +1)*(x^2 +x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Aug 07 2013

A103947 a(n) is the number of distinct n-th powers of functions {1, 2} -> {1, 2}.

Original entry on oeis.org

1, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3

Offset: 0

David Wasserman, Feb 21 2005

			a(4) = 3: the four functions {1, 2} -> {1, 2} are f(x) = 1, g(x) = 2, h(x) = x and j(x) = 3 - x. f^4(x) = f(f(f(f(x)))) = 1; so f^4 = f. Similarly, g^4 = g, h^4 = h and j^4 = h, so there are 3 distinct 4th powers.

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A102687, A102709, A103948, A103949, A103950.
Cf. A158515.
Row n=2 of A247026.

Join[{1},LinearRecurrence[{0, 1},{4, 3},104]] (* Ray Chandler, Sep 08 2015 *)

For n > 2, a(n) = a(n-2).
G.f.: (1+4*x+2*x^2)/(1-x^2). - Jaume Oliver Lafont, Mar 20 2009
a(n) = (n mod 2)+(2 mod (n+2))+1. - Aaron J Grech, Sep 02 2024
E.g.f.: 3*cosh(x) + 4*sinh(x) - 2. - Stefano Spezia, Sep 04 2024

A103949 a(n) is the number of distinct n-th powers of functions {1, 2, 3, 4} -> {1, 2, 3, 4}.

Original entry on oeis.org

1, 256, 100, 116, 73, 148, 44, 148, 73, 116, 76, 148, 41, 148, 76, 116, 73, 148, 44, 148, 73, 116, 76, 148, 41, 148, 76, 116, 73, 148, 44, 148, 73, 116, 76, 148, 41, 148, 76, 116, 73, 148, 44, 148, 73, 116, 76, 148, 41, 148, 76, 116, 73, 148, 44, 148, 73, 116, 76, 148

Offset: 0

David Wasserman, Feb 21 2005

For n > 2, a(n) = a(n+12).

Index entries for linear recurrences with constant coefficients, signature (-1, -1, 0, 1, 1, 1).

Cf. A102687, A102709, A103947, A103948, A103950.

Row n=4 of A247026.

LinearRecurrence[{-1,-1,0,1,1,1},{1,256,100,116,73,148,44,148,73},60] (* Harvey P. Dale, Jun 22 2015 *)

G.f.: (24*x^8+132*x^7+92*x^6-80*x^5-288*x^4-472*x^3-357*x^2-257*x-1) / ((x-1)*(x+1)*(x^2+1)*(x^2+x+1)). - Colin Barker, Aug 07 2013

A103948 a(n) is the number of distinct n-th powers of functions {1, 2, 3} -> {1, 2, 3}.

Original entry on oeis.org

1, 27, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19

Offset: 0

David Wasserman, Feb 21 2005

Index entries for linear recurrences with constant coefficients, signature (-1, 0, 1, 1).

Cf. A102687, A102709, A103947, A103949, A103950.

Row n=3 of A247026.

Join[{1, 27},LinearRecurrence[{-1, 0, 1, 1},{12, 19, 12, 21},68]] (* Ray Chandler, Sep 08 2015 *)

For n > 1, a(n) = a(n+6).

G.f.: (6*x^5-3*x^4-30*x^3-39*x^2-28*x-1) / ((x-1)*(x+1)*(x^2+x+1)). - Colin Barker, Aug 07 2013

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.

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A102687 Number of different squares of labeled mappings of a finite set of n elements into itself.

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A103950 a(n) is the number of distinct n-th powers of functions {1, 2, 3, 4, 5, 6} -> {1, 2, 3, 4, 5, 6}.

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A103947 a(n) is the number of distinct n-th powers of functions {1, 2} -> {1, 2}.

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A103949 a(n) is the number of distinct n-th powers of functions {1, 2, 3, 4} -> {1, 2, 3, 4}.

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A103948 a(n) is the number of distinct n-th powers of functions {1, 2, 3} -> {1, 2, 3}.

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