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 A356887 Number of n X n matrices over GF(2) whose characteristic polynomial is a single monic irreducible (prime) raised to some power. 0
 1, 2, 10, 176, 14016, 4032512, 6213763072, 32018926665728, 870713558978002944, 89293629194528350011392, 40675925233031615853327548416, 72389802739964734146185851566030848, 563250609270594469597103043401725627072512 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 11 16:53 EDT 2024. Contains 375073 sequences. (Running on oeis4.)