%I #9 Jul 21 2021 09:29:03

%S 1,1,3,1,21,42,1,105,1470,2520,1,465,32550,390600,624960,1,1953,

%T 605430,36325800,406848960,629959680,1,8001,10417302,2768025960,

%U 155009453760,1680102466560,2560156139520,1,32385,172741590,192779614440,47809344381120,2590958018073600,27636885526118400,41781748196966400

%N Triangular array read by rows: T(n,k) is the number of nilpotent n X n matrices over GF(2) having rank k, 0 <= k <= n-1, n >= 1.

%H 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 Array begins

%e 1;

%e 1, 3;

%e 1, 21, 42;

%e 1, 105, 1470, 2520;

%e 1, 465, 32550, 390600, 624960;

%e 1, 1953, 605430, 36325800, 406848960, 629959680

%e T(2,0) = 1 because the zero matrix has rank 0.

%e T(2,1) = 3 because {{0,0},{1,0}}, {{0,1},{0,0}}, {{1,1},{1,1}} have rank 1.

%t nn = 10; 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[

%t 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}];

%t g[u_, v_] := Total[Map[v^(Total[#] - Length[#]) u^Total[#]/aut[1, #] &,

%t Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]];Map[Select[#, # > 0 &] &,Drop[Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[

%t Series[g[u, v], {u, 0, nn}], {u, v}], 1]] // Grid

%Y Cf. A134057 (column k=1), A083402 (main diagonal), A053763 (row sums).

%K nonn,tabl

%O 1,3

%A _Geoffrey Critzer_, Jul 15 2021