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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.

%I A346743 #24 Jun 30 2022 14:45:04
%S A346743 1,1,3,2,1,21,56,42,0,0,48,1,315,1232,3780,1344,5040,5760,0,0,0,0,0,0,
%T A346743 0,2688,1,6975,75392,416640,666624,1249920,476160,624960,0,0,0,833280,
%U A346743 0,1428480,1333248,0,0,0,0,0,952320,0,0,0,0,0,0,0,0,0,1935360
%N 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.
%H A346743 M. R. Darafsheh, <a href="https://doi.org/10.1016/j.ffa.2004.12.003">Order of elements in the groups related to the general linear group</a>, Finite fields and their applications, 11 (2005), 738-747.
%H A346743 Joseph Kung, <a href="https://doi.org/10.1016/0024-3795(81)90227-5">The Cycle Structure of a Linear Transformation over a Finite Field</a>, Linear Algebra and its Applications, Vol 36, 1981, pages 141-155.
%H A346743 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%F A346743 T(n,2^n - 1) = A346019(n).
%e A346743   1,
%e A346743   1,   3,    2,
%e A346743   1,  21,   56,   42,    0,    0,   48,
%e A346743   1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688
%t A346743 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, #] &,
%t A346743   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}]
%Y A346743 Cf. A002884 (row sums), A346019.
%K A346743 nonn,tabf
%O A346743 1,3
%A A346743 _Geoffrey Critzer_, Jul 31 2021