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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.

%I A346421 #8 Jul 21 2021 09:29:57
%S A346421 1,0,3,18,24,39,3456,8190,5928,2109,7619508,17094240,13700700,4215120,
%T A346421 417153,149200289280,335730157884,267485755680,85615372260,8910314160,
%U A346421 346720179,26394940582090344,59388527912287392,47325384827973252,15262273318168800,1648005959253654,74268805562952,1233891662727
%N 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.
%H A346421 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 A346421              1;
%e A346421              0,            3;
%e A346421             18,           24,           39;
%e A346421           3456,         8190,         5928,        2109;
%e A346421        7619508,     17094240,     13700700,     4215120,     417153;
%e A346421   149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179;
%t A346421 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 =
%t A346421 Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; g[u_, v_] :=
%t A346421 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
%Y A346421 Cf. A346209 (column k=0), A290516 (main diagonal), A060722 (row sums).
%K A346421 nonn,tabl
%O A346421 0,3
%A A346421 _Geoffrey Critzer_, Jul 16 2021