A171806 Number of 5 X 5 permutation matrices such that the n-th matrix power is the least nonnegative power that gives the identity matrix.

1, 25, 20, 30, 24, 20

Offset

1,2

Comments

The sum of the terms of this sequence is equal to the number of 5 X 5 permutation matrices: 5! = 120.

Number of elements of order n in symmetric group S_5. - Alois P. Heinz, Mar 30 2020

Links

Table of n, a(n) for n=1..6.

Example

a(1) = 1 because there is only one matrix whose first power is the identity matrix (this is the identity matrix itself).

Mathematica

tab = {0, 0, 0, 0, 0, 0}; per =
 Permutations[{1, 2, 3, 4, 5}]; zeromat = {}; Do[
 AppendTo[zeromat, Table[0, {n, 1, 5}]], {m, 1, 5}]; unit =
 IdentityMatrix[5]; s5 = {}; Do[s = zeromat;
 Do[s[[m]][[per[[n]][[m]]]] = 1, {m, 1, 5}];
 AppendTo[s5, s], {n, 1, 120}]; Do[
 If[MatrixPower[s5[[n]], 1] == unit, tab[[1]] = tab[[1]] + 1,
  If[MatrixPower[s5[[n]], 2] == unit, tab[[2]] = tab[[2]] + 1,
   If[MatrixPower[s5[[n]], 3] == unit, tab[[3]] = tab[[3]] + 1,
    If[MatrixPower[s5[[n]], 4] == unit, tab[[4]] = tab[[4]] + 1,
     If[MatrixPower[s5[[n]], 5] == unit, tab[[5]] = tab[[5]] + 1,
      If[MatrixPower[s5[[n]], 6] == unit,
       tab[[6]] = tab[[6]] + 1]]]]]], {n, 1, 120}]; tab

Crossrefs

Row n=5 of A057731.

Sequence in context: A104790 A291429 A334562 * A038822 A375335 A061438

Adjacent sequences: A171803 A171804 A171805 * A171807 A171808 A171809

Keyword

nonn,fini,full

Author

Artur Jasinski, Dec 18 2009

Extensions

Name edited and terms corrected by Alois P. Heinz, Mar 30 2020

Status

approved