A171806 - OEIS

%I #22 May 11 2020 01:38:54

%S 1,25,20,30,24,20

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

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

%C Number of elements of order n in symmetric group S_5. - _Alois P. Heinz_, Mar 30 2020

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

%t tab = {0, 0, 0, 0, 0, 0}; per =

%t  Permutations[{1, 2, 3, 4, 5}]; zeromat = {}; Do[

%t  AppendTo[zeromat, Table[0, {n, 1, 5}]], {m, 1, 5}]; unit =

%t  IdentityMatrix[5]; s5 = {}; Do[s = zeromat;

%t  Do[s[[m]][[per[[n]][[m]]]] = 1, {m, 1, 5}];

%t  AppendTo[s5, s], {n, 1, 120}]; Do[

%t  If[MatrixPower[s5[[n]], 1] == unit, tab[[1]] = tab[[1]] + 1,

%t   If[MatrixPower[s5[[n]], 2] == unit, tab[[2]] = tab[[2]] + 1,

%t    If[MatrixPower[s5[[n]], 3] == unit, tab[[3]] = tab[[3]] + 1,

%t     If[MatrixPower[s5[[n]], 4] == unit, tab[[4]] = tab[[4]] + 1,

%t      If[MatrixPower[s5[[n]], 5] == unit, tab[[5]] = tab[[5]] + 1,

%t       If[MatrixPower[s5[[n]], 6] == unit,

%t        tab[[6]] = tab[[6]] + 1]]]]]], {n, 1, 120}]; tab

%Y Row n=5 of A057731.

%K nonn,fini,full

%O 1,2

%A _Artur Jasinski_, Dec 18 2009

%E Name edited and terms corrected by _Alois P. Heinz_, Mar 30 2020