A347010 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) with minimal polynomial of degree k, n >= 0, 0 <= k <= n.

1, 0, 2, 0, 2, 14, 0, 2, 98, 412, 0, 2, 1542, 13160, 50832, 0, 2, 34782, 1147744, 6854720, 25517184, 0, 2, 1908734, 260411904, 2544075264, 14153094144, 51759986688, 0, 2, 166738046, 107691724672, 2985421682688, 21570911944704, 116285097148416, 422000664182784

Offset

0,3

Links

Table of n, a(n) for n=0..35.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Example

  1,
  0, 2,
  0, 2,   14,
  0, 2,   98,   412,
  0, 2, 1542, 13160, 50832

Mathematica

nn = 8; 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]}]; \[Nu] =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; L = Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]; g[u_, v_, deg_] :=
Total[Map[v^(Max[Prepend[#, 0]] deg) u^(deg Total[#])/aut[deg, #] &,
   L]]; Table[Take[(Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[g[u, v, deg]^\[Nu][[deg]], {deg, 1, nn}], {u, 0, nn}], {u, v}])[[n]], n], {n, 1, nn}] // Grid

Crossrefs

Cf. A002416 (row sums), A346082 (main diagonal).

Sequence in context: A230813 A367074 A177113 * A281205 A285152 A077184

Adjacent sequences: A347007 A347008 A347009 * A347011 A347012 A347013

Keyword

nonn,tabl

Author

Geoffrey Critzer, Aug 10 2021

Status

approved