A346384 Triangle read by rows. T(n,k) is the number of invertible n X n matrices over GF(3) such that the dimension of the eigenspace corresponding to the eigenvalue 1 is k, 0 <= k <= n, n >= 0.

1, 1, 1, 27, 20, 1, 6291, 4719, 221, 1, 13589289, 10191960, 477750, 2120, 1, 266377183929, 199782888129, 9364822830, 41559870, 19481, 1, 47123189360124723, 35342392020078780, 1656674625945339, 7352106327720, 3446299857, 176540, 1

Offset: 0

Author

Geoffrey Critzer, Jul 14 2021

Keywords

• nonn
• tabl

Examples

			             1;
             1,            1;
            27,           20,          1;
          6291,         4719,        221,        1;
      13589289,     10191960,     477750,     2120,     1;
  266377183929, 199782888129, 9364822830, 41559870, 19481, 1;
		

Links

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

Crossrefs

Cf. A051680 (column k=0), A053290 (row sums).

Programs

Mathematica

nn = 6; 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]}]; A027376 = 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}]]]; Map[Select[#, # > 0 &] &, Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0,  nn}] CoefficientList[
    Series[(g[u, v] /. v -> 1)*g[u, v]* Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A027376[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}]] // Grid