A356887 Number of n X n matrices over GF(2) whose characteristic polynomial is a single monic irreducible (prime) raised to some power.

1, 2, 10, 176, 14016, 4032512, 6213763072, 32018926665728, 870713558978002944, 89293629194528350011392, 40675925233031615853327548416, 72389802739964734146185851566030848, 563250609270594469597103043401725627072512

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..12.

Wikipedia, Frobenius normal form

Mathematica

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[
  Series[1 + Sum[\[Nu][[d]]*(g[u, 1, d] - 1), {d, 1, nn}] , {u, 0, nn}], u]

Crossrefs

Sequence in context: A069994 A063573 A368573 * A086675 A319607 A057119

Adjacent sequences: A356884 A356885 A356886 * A356888 A356889 A356890

Keyword

nonn

Author

Geoffrey Critzer, Sep 02 2022

Status

approved