A346201 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) such that the sum of the dimensions of their eigenspaces taken over all eigenvalues is k, 0 <= k <= n, n >= 0.

1, 0, 2, 2, 6, 8, 48, 196, 210, 58, 5824, 23280, 27020, 8610, 802, 2887680, 11550848, 13756560, 4757260, 581250, 20834, 5821595648, 23286380544, 28097284992, 10075582800, 1369706604, 67874562, 1051586, 47317927329792, 189271709384704, 229853403924480, 83865929653632, 11957394226896, 668707460652, 14779207170, 102233986

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;
        2,        6,        8;
       48,      196,      210,      58;
     5824,    23280,    27020,    8610,    802;
  2887680, 11550848, 13756560, 4757260, 581250, 20834;

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]}]; A001037 =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; g[u_, v_] :=
Total[Map[v^Length[#] u^Total[#]/aut[1, #] &, 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[g[u, v] g[u, v] Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A001037[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}])[[n]],
   n], {n, 1, nn}] // Grid

Crossrefs

Cf. A002820 (column k=0), A132186 (main diagonal), A002416 (row sums).

Sequence in context: A334499 A357950 A201499 * A131553 A277510 A169800

Adjacent sequences: A346198 A346199 A346200 * A346202 A346203 A346204

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jul 16 2021

Status

approved