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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

%I #11 Aug 24 2021 06:49:06

%S 1,0,2,0,2,14,0,2,98,412,0,2,1542,13160,50832,0,2,34782,1147744,

%T 6854720,25517184,0,2,1908734,260411904,2544075264,14153094144,

%U 51759986688,0,2,166738046,107691724672,2985421682688,21570911944704,116285097148416,422000664182784

%N 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.

%H 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 1,

%e 0, 2,

%e 0, 2, 14,

%e 0, 2, 98, 412,

%e 0, 2, 1542, 13160, 50832

%t 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] =

%t 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_] :=

%t Total[Map[v^(Max[Prepend[#, 0]] deg) u^(deg Total[#])/aut[deg, #] &,

%t 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

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

%K nonn,tabl

%O 0,3

%A _Geoffrey Critzer_, Aug 10 2021

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Last modified March 29 02:22 EDT 2024. Contains 371264 sequences. (Running on oeis4.)