%I #7 Sep 03 2022 08:47:31
%S 1,2,10,176,14016,4032512,6213763072,32018926665728,
%T 870713558978002944,89293629194528350011392,
%U 40675925233031615853327548416,72389802739964734146185851566030848,563250609270594469597103043401725627072512
%N Number of n X n matrices over GF(2) whose characteristic polynomial is a single monic irreducible (prime) raised to some power.
%C Equivalently, a(n) is the number of n X n matrices over GF(2) whose invariant factors are the same as its elementary divisors. In other words, the matrices whose rational canonical form is equal to the primary rational canonical form.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Frobenius_normal form">Frobenius normal form</a>
%t nn = 12; q = 2; b[p_, i_] := Count[p, i];s[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^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}],n,1,nn}]; l[greatestpart_] := Level[Table[ IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0,nn}], {2}];g[u_, v_, deg_] := Total[Map[v^Total[#] u^(deg Total[#])/aut[deg, #] &, l[nn]]];Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
%t Series[1 + Sum[\[Nu][[d]]*(g[u, 1, d] - 1), {d, 1, nn}] , {u, 0,nn}], u]
%K nonn
%O 0,2
%A _Geoffrey Critzer_, Sep 02 2022