A346743 Irregular triangular array read by rows. T(n,k) is the number of matrices in GL_n(F_2) having order k, 1<=k<=2^n-1, n>=1.

1, 1, 3, 2, 1, 21, 56, 42, 0, 0, 48, 1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688, 1, 6975, 75392, 416640, 666624, 1249920, 476160, 624960, 0, 0, 0, 833280, 0, 1428480, 1333248, 0, 0, 0, 0, 0, 952320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1935360

Offset: 1

Author

Geoffrey Critzer, Jul 31 2021

Keywords

• nonn
• tabf

Examples

			  1,
  1,   3,    2,
  1,  21,   56,   42,    0,    0,   48,
  1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688
		

Links

• M. R. Darafsheh, Order of elements in the groups related to the general linear group, Finite fields and their applications, 11 (2005), 738-747.
• 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.

Crossrefs

Cf. A002884 (row sums), A346019.

Programs

Mathematica

nn = 7; 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}]; Table[a = Drop[Transpose[  Table[g[u_, v_, deg_] :=Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &,
  Level[Table[IntegerPartitions[n, {0, n}, Range[Drop[FactorList[z^k - 1, Modulus -> q], 1][[1,2]]]], {n, 0, nn}], {2}]]];degreelist =Map[Exponent[#, z] &, Drop[FactorList[z^k - 1, Modulus -> q], 1][[All, 1]]];Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0,nn}] CoefficientList[Series[Product[g[u, 1, deg], {deg, degreelist}], {u, 0, nn}],u], {k, 1, 2^nn - 1}]], 1][[n]];Nest[Append[#,   a[[Length[#] + 1]] - Sum[#[[j]], {j, Drop[Divisors[Length[#] + 1], -1]}]] & , {1},2^n - 2], {n, 1, nn}]

Formula

T(n,2^n - 1) = A346019(n).