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A346412 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.

%I A346412 #20 Dec 23 2024 22:06:51
%S A346412 1,1,3,1,21,42,1,105,1470,2520,1,465,32550,390600,624960,1,1953,
%T A346412 605430,36325800,406848960,629959680,1,8001,10417302,2768025960,
%U A346412 155009453760,1680102466560,2560156139520,1,32385,172741590,192779614440,47809344381120,2590958018073600,27636885526118400,41781748196966400
%N A346412 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.
%D A346412 G. Lusztig, A note on counting nilpotent matrices of fixed rank, Bull. London Math. Soc. v.8  (1976), no. 1, 77--80; MR0407050.
%H A346412 Jason Fulman, <a href="https://arxiv.org/abs/math/9712238">A probabilistic approach toward the finite general linear and unitary group</a>, arxiv.org, Dec 1997, page 12.
%H A346412 J. Kung, <a href="https://doi.org/10.1016/0024-3795(81)90227-5">The cycle structure of a linear transformation over a finite field</a>,Linear Algebra and its Applications, vol 36, March 1981,141-155.
%H A346412 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.
%F A346412 T(n,n-k) = A002884(n)*Product_{i=k..n-1}(1-1/2^i)/(A002884(k)*2^(n-k)*Product_{i=1..n-k}(1-1/2^i)) Theorem 6 in Fulman link. - _Geoffrey Critzer_, Dec 23 2024
%e A346412 Array begins
%e A346412   1;
%e A346412   1,    3;
%e A346412   1,   21,     42;
%e A346412   1,  105,   1470,     2520;
%e A346412   1,  465,  32550,   390600,    624960;
%e A346412   1, 1953, 605430, 36325800, 406848960, 629959680
%e A346412 T(2,0) = 1 because the zero matrix has rank 0.
%e A346412 T(2,1) = 3 because {{0,0},{1,0}}, {{0,1},{0,0}}, {{1,1},{1,1}} have rank 1.
%t A346412 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 A346412    q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}];
%t A346412 g[u_, v_] := Total[Map[v^(Total[#] - Length[#]) u^Total[#]/aut[1, #] &,
%t A346412    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 A346412      Series[g[u, v], {u, 0, nn}], {u, v}], 1]] // Grid
%Y A346412 Cf. A134057 (column k=1), A083402 (main diagonal), A053763 (row sums).
%K A346412 nonn,tabl
%O A346412 1,3
%A A346412 _Geoffrey Critzer_, Jul 15 2021