A385355 Triangular array read by rows: T(n,k) is the number of n X n matrices A over GF(2) such that the dimension of the null space of A^n is equal to k, n>=0, 0<=k<=n.

1, 1, 1, 6, 6, 4, 168, 168, 112, 64, 20160, 20160, 13440, 7680, 4096, 9999360, 9999360, 6666240, 3809280, 2031616, 1048576, 20158709760, 20158709760, 13439139840, 7679508480, 4095737856, 2113929216, 1073741824, 163849992929280, 163849992929280, 109233328619520, 62419044925440, 33290157293568, 17182016667648, 8727373545472, 4398046511104

Offset: 0

Author

Geoffrey Critzer, Jun 26 2025

Keywords

• nonn
• tabl

Comments

Conjecture: In the limit as n goes to infinity the probability that the nullity of such a random matrix is equal to k is Product_{i>=1} (1-1/2^i * 2^binomial(k,2)/A005329(k)).

Examples

			Triangle T(n,k) begins:
        1;
        1,       1;
        6,       6,       4;
      168,     168,     112,      64;
    20160,   20160,   13440,    7680,    4096;
  9999360, 9999360, 6666240, 3809280, 2031616, 1048576;
  ...
		

Crossrefs

Cf. A002884 (column k=0), A053763 (main diagonal), A002416 (row sums), A005329.

Programs

Mathematica

nn = 6; 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]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; 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^Total[#] u^(deg Total[#])/aut[deg, #] &, partitions]];Map[Select[#, # > 0 &] &,Table[\[Gamma][n, q], {n, 0, nn}] CoefficientList[Series[g[u, v, 1, l]*g[u, 1, 1, l] Product[g[u, 1, deg, l]^\[Nu][[deg]], {deg, 2, nn}], {u, 0, nn}], {u,v}]] // Grid