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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.

%I A385355 #18 Jul 01 2025 19:36:31
%S A385355 1,1,1,6,6,4,168,168,112,64,20160,20160,13440,7680,4096,9999360,
%T A385355 9999360,6666240,3809280,2031616,1048576,20158709760,20158709760,
%U A385355 13439139840,7679508480,4095737856,2113929216,1073741824,163849992929280,163849992929280,109233328619520,62419044925440,33290157293568,17182016667648,8727373545472,4398046511104
%N 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.
%C A385355 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)).
%e A385355 Triangle T(n,k) begins:
%e A385355         1;
%e A385355         1,       1;
%e A385355         6,       6,       4;
%e A385355       168,     168,     112,      64;
%e A385355     20160,   20160,   13440,    7680,    4096;
%e A385355   9999360, 9999360, 6666240, 3809280, 2031616, 1048576;
%e A385355   ...
%t A385355 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
%Y A385355 Cf. A002884 (column k=0), A053763 (main diagonal), A002416 (row sums), A005329.
%K A385355 nonn,tabl
%O A385355 0,4
%A A385355 _Geoffrey Critzer_, Jun 26 2025