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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.

%I A346214 #31 Jun 11 2025 10:09:26
%S A346214 1,1,3,1,21,42,1,315,1260,2520,1,6975,104160,312480,624960,1,373023,
%T A346214 23436000,104993280,314979840,629959680,1,32252031,9175162752,
%U A346214 121912197120,426692689920,1280078069760,2560156139520,1,6619979775,9978120069120,421755245936640,1989607056998400,6963624699494400,20890874098483200,41781748196966400
%N 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.
%C A346214 The index of a nilpotent matrix A is the smallest positive integer k such that A^k = 0.
%C A346214 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
%e A346214   1,
%e A346214   1,    3,
%e A346214   1,   21,     42,
%e A346214   1,  315,   1260,   2520,
%e A346214   1, 6975, 104160, 312480, 624960
%t A346214 nn = 8;  q = 2;
%t A346214 b[p_, i_] := Count[p, i];
%t A346214 d[p_, i_] :=  Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];
%t A346214 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]}];
%t A346214 l = Level[Table[IntegerPartitions[n],  {n,  0,  nn}],  {2}];
%t A346214 \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}];
%t A346214 g[u_,  v_,  deg_,  partitions_] := Total[Map[v^If[# == {},  0,  Max[#]] u^(deg Total[#])/aut[deg,  #] &, partitions]];
%t A346214 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
%Y A346214 Cf. A083402 (main diagonal), A053763 (row sums), A002884, A048651.
%K A346214 nonn,tabl
%O A346214 1,3
%A A346214 _Geoffrey Critzer_, Jul 10 2021
%E A346214 More terms from _Geoffrey Critzer_, Jun 10 2025