A346214 Triangular array read by rows. T(n,k) is the number of nilpotent n X n matrices over GF(2) with index k, 1 <= k <= n, n >= 1.

1, 1, 3, 1, 21, 42, 1, 315, 1260, 2520, 1, 6975, 104160, 312480, 624960, 1, 373023, 23436000, 104993280, 314979840, 629959680, 1, 32252031, 9175162752, 121912197120, 426692689920, 1280078069760, 2560156139520, 1, 6619979775, 9978120069120, 421755245936640, 1989607056998400, 6963624699494400, 20890874098483200, 41781748196966400

Offset: 1

Author

Geoffrey Critzer, Jul 10 2021

Keywords

• nonn
• tabl

Comments

The index of a nilpotent matrix A is the smallest positive integer k such that A^k = 0.

Define the co-index of an n X n matrix A to be n - index(A). Let X_n be the random variable that assigns to each nilpotent n X n matrix over GF(2) the value j in {0,1,...,n-1} of its co-index. Conjecture: lim_{n->inf} P(X_n = j) = Product_{i>=1}1-1/2^i * 2^((j-1)^2)/A002884(j). Moreover, for j < 2n, T(n,n-j) = A002884(n)/(A002884(j)*2^(n - (j-1)^2)). - Geoffrey Critzer, Jun 10 2025

Examples

			  1,
  1,    3,
  1,   21,     42,
  1,  315,   1260,   2520,
  1, 6975, 104160, 312480, 624960
		

Crossrefs

Cf. A083402 (main diagonal), A053763 (row sums), A002884, A048651.

Programs

Mathematica

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]}];
l = Level[Table[IntegerPartitions[n],  {n,  0,  nn}],  {2}];
\[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}];
g[u_,  v_,  deg_,  partitions_] := Total[Map[v^If[# == {},  0,  Max[#]] u^(deg Total[#])/aut[deg,  #] &, partitions]];
Map[Select[#,  # > 0 &] &, Drop[Table[\[Gamma][n,  q],  {n,  0,  nn}] CoefficientList[     Series[g[u,  v,  1,  l],  {u,  0,  nn}],  {u,  v}],  1]] // Grid

Extensions

More terms from Geoffrey Critzer, Jun 10 2025