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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.

%I A346201 #21 Nov 15 2022 12:56:19
%S A346201 1,0,2,2,6,8,48,196,210,58,5824,23280,27020,8610,802,2887680,11550848,
%T A346201 13756560,4757260,581250,20834,5821595648,23286380544,28097284992,
%U A346201 10075582800,1369706604,67874562,1051586,47317927329792,189271709384704,229853403924480,83865929653632,11957394226896,668707460652,14779207170,102233986
%N 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.
%H A346201 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 A346201         1;
%e A346201         0,        2;
%e A346201         2,        6,        8;
%e A346201        48,      196,      210,      58;
%e A346201      5824,    23280,    27020,    8610,    802;
%e A346201   2887680, 11550848, 13756560, 4757260, 581250, 20834;
%t A346201 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 =
%t A346201 Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];g[u_, v_] :=
%t A346201 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]],
%t A346201    n], {n, 1, nn}] // Grid
%Y A346201 Cf. A002820 (column k=0), A132186 (main diagonal), A002416 (row sums).
%K A346201 nonn,tabl
%O A346201 0,3
%A A346201 _Geoffrey Critzer_, Jul 16 2021