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.

1, 1, 3, 1, 21, 42, 1, 105, 1470, 2520, 1, 465, 32550, 390600, 624960, 1, 1953, 605430, 36325800, 406848960, 629959680, 1, 8001, 10417302, 2768025960, 155009453760, 1680102466560, 2560156139520, 1, 32385, 172741590, 192779614440, 47809344381120, 2590958018073600, 27636885526118400, 41781748196966400

Offset

1,3

Links

Table of n, a(n) for n=1..36.

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

Example

Array begins
  1;
  1,    3;
  1,   21,     42;
  1,  105,   1470,     2520;
  1,  465,  32550,   390600,    624960;
  1, 1953, 605430, 36325800, 406848960, 629959680
T(2,0) = 1 because the zero matrix has rank 0.
T(2,1) = 3 because {{0,0},{1,0}}, {{0,1},{0,0}}, {{1,1},{1,1}} have rank 1.

Mathematica

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[
   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}];
g[u_, v_] := Total[Map[v^(Total[#] - Length[#]) u^Total[#]/aut[1, #] &,
   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[
     Series[g[u, v], {u, 0, nn}], {u, v}], 1]] // Grid

Crossrefs

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

Sequence in context: A223549 A138354 A193632 * A346214 A190962 A010291

Adjacent sequences: A346409 A346410 A346411 * A346413 A346414 A346415

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jul 15 2021

Status

approved