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A346677 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) that can be decomposed as the direct sum of k cyclic matrices, 0<=k<=n, n>=0. 0
1, 0, 2, 0, 8, 8, 0, 132, 322, 58, 0, 10752, 36412, 17570, 802, 0, 3185280, 16923024, 11693324, 1731970, 20834, 0, 5279662080, 26989750656, 30003846992, 6109974636, 335190786, 1051586, 0, 28343145922560, 196717668747264, 247267921788288, 84586214764240, 5906325116460, 128574848514, 102233986 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

Joseph Kung, The Cycle Structure of a Linear Transformation over a Finite Field, Linear Algebra and its Applications, Vol 36, 1981, pages 141-155.

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

EXAMPLE

Triangle begins:

1;

0, 2;

0, 8, 8;

0, 132, 322, 58;

0, 10752, 36412, 17570, 802;

0, 3185280, 16923024, 11693324, 1731970, 20834;

...

MATHEMATICA

nn = 6; 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]}];

A001037 = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];

g[u_, v_, deg_] := Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &, Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]];

Table[Take[(Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[g[u, v, deg]^A001037[[deg]], {deg, 1, nn}], {u, 0, nn}], {u, v}])[[n]], n], {n, 1, nn}] // Grid

CROSSREFS

Cf. A002416 (row sums) A132186 (main diagonal).

Sequence in context: A288015 A288438 A287737 * A240340 A288700 A287855

Adjacent sequences: A346674 A346675 A346676 * A346678 A346679 A346680

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Jul 28 2021

STATUS

approved

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Last modified February 7 02:40 EST 2023. Contains 360111 sequences. (Running on oeis4.)