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

1,3

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

Links

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

Example

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

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

Crossrefs

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

Sequence in context: A138354 A193632 A346412 * A190962 A010291 A372167

Adjacent sequences: A346211 A346212 A346213 * A346215 A346216 A346217

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jul 10 2021

Extensions

More terms from Geoffrey Critzer, Jun 10 2025

Status

approved