A346421 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(3) such that the sum of the dimensions of its eigenspaces taken over all its eigenvalues is k, 0 <= k <= n, n >= 0.

1, 0, 3, 18, 24, 39, 3456, 8190, 5928, 2109, 7619508, 17094240, 13700700, 4215120, 417153, 149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179, 26394940582090344, 59388527912287392, 47325384827973252, 15262273318168800, 1648005959253654, 74268805562952, 1233891662727

Offset

0,3

Links

Table of n, a(n) for n=0..27.

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

Example

             1;
             0,            3;
            18,           24,           39;
          3456,         8190,         5928,        2109;
       7619508,     17094240,     13700700,     4215120,     417153;
  149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179;

Mathematica

nn = 7; q = 3; 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]^3 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. A346209 (column k=0), A290516 (main diagonal), A060722 (row sums).

Sequence in context: A174029 A103715 A131860 * A263578 A048080 A306279

Adjacent sequences: A346418 A346419 A346420 * A346422 A346423 A346424

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jul 16 2021

Status

approved