A346081 Triangular array read by rows. T(n,k) is the number of invertible n X n matrices over GF(2) that decompose into the direct sum of k cyclic matrices, 0<=k<=n, n>=0.

1, 0, 1, 0, 5, 1, 0, 90, 77, 1, 0, 8232, 10702, 1225, 1, 0, 2560320, 6437832, 980902, 20305, 1, 0, 4649702400, 11245947840, 4174172072, 88552182, 335265, 1, 0, 25782989783040, 94915404702720, 39725105598144, 3416196663656, 10290711702, 5470017, 1

Offset

0,5

Links

Table of n, a(n) for n=0..35.

Joseph Kung, The Cycle Structure of a Linear Transformation over a Finite Field, Linear Algebra and its Applications, Vol 36, 1981, pages 141-155.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Example

  1;
  0,       1;
  0,       5,       1;
  0,      90,      77,      1;
  0,    8232,   10702,   1225,     1;
  0, 2560320, 6437832, 980902, 20305, 1;

Mathematica

nn = 8; q = 2; b[p_, i_] := Count[p, i]; d[p_, i_] :=Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}]; aut[deg_, p_] :=Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}]; A001037 =Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; g[u_, v_, deg_] := Total[Map[v^Length[#]u^(deg Total[#])/aut[deg, #]&, Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]]; Table[Take[(Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g[u, v, 1] Product[g[u, v, deg]^A001037[[deg]], {deg, 2, nn}], {u, 0, nn}], {u, v}])[[n]], n], {n, 1, nn}] // Grid

Crossrefs

Cf. A002884 (row sums).

Sequence in context: A256042 A292604 A112991 * A137373 A220962 A201292

Adjacent sequences: A346078 A346079 A346080 * A346082 A346083 A346084

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jul 29 2021

Status

approved