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A346677 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) that can be decomposed as the direct sum of k cyclic matrices, 0<=k<=n, n>=0.

%I A346677 #19 Jun 30 2022 12:40:25
%S A346677 1,0,2,0,8,8,0,132,322,58,0,10752,36412,17570,802,0,3185280,16923024,
%T A346677 11693324,1731970,20834,0,5279662080,26989750656,30003846992,
%U A346677 6109974636,335190786,1051586,0,28343145922560,196717668747264,247267921788288,84586214764240,5906325116460,128574848514,102233986
%N A346677 Triangular array read by rows.  T(n,k) is the number of n X n matrices over GF(2) that can be decomposed as the direct sum of k cyclic matrices, 0<=k<=n, n>=0.
%H A346677 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 A346677 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.
%e A346677 Triangle begins:
%e A346677   1;
%e A346677   0,       2;
%e A346677   0,       8,        8;
%e A346677   0,     132,      322,       58;
%e A346677   0,   10752,    36412,    17570,     802;
%e A346677   0, 3185280, 16923024, 11693324, 1731970, 20834;
%e A346677   ...
%t A346677 nn = 6; q = 2; b[p_, i_] := Count[p, i];
%t A346677 d[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];
%t A346677 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]}];
%t A346677 A001037 = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];
%t A346677 g[u_, v_, deg_] :=  Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &, Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]];
%t A346677 Table[Take[(Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[g[u, v, deg]^A001037[[deg]], {deg, 1, nn}], {u, 0, nn}], {u, v}])[[n]], n], {n, 1, nn}] // Grid
%Y A346677 Cf. A002416 (row sums) A132186 (main diagonal).
%K A346677 nonn,tabl
%O A346677 0,3
%A A346677 _Geoffrey Critzer_, Jul 28 2021