A348622 Triangular array read by rows: T(n,k) is the number of periodic n X n matrices over GF(2) having rank k, n>=0, 0<=k<=n.

1, 1, 1, 1, 6, 6, 1, 28, 168, 168, 1, 120, 3360, 20160, 20160, 1, 496, 59520, 1666560, 9999360, 9999360, 1, 2016, 999936, 119992320, 3359784960, 20158709760, 20158709760, 1, 8128, 16386048, 8127479808, 975297576960, 27308332154880, 163849992929280, 163849992929280

Offset: 0

Author

Geoffrey Critzer, Oct 25 2021

Keywords

• nonn
• tabl

Comments

A matrix T is periodic if and only image(T) = image(T^2). Cf. A348015.

Examples

			Triangle begins:
  1;
  1,   1;
  1,   6,     6;
  1,  28,   168,     168;
  1, 120,  3360,   20160,   20160;
  1, 496, 59520, 1666560, 9999360, 9999360;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..57, flattened
• Eric Weisstein's World of Mathematics, Periodic Matrix

Crossrefs

Cf. A348015 (row sums).

Main diagonal gives A002884.

Programs

Maple

b:= proc(n) option remember; mul(2^n-2^i, i=0..n-1) end:
T:= (n, k)-> b(n)/b(n-k):
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Oct 30 2021

Mathematica

nn = 7; q = 2; b[p_, i_] := Count[p, i];s[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^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}], {n, 1,nn}]; l[greatestpart_]:=Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0, nn}], {2}];
g1[u_, v_, deg_] :=Total[Map[v^(Length[#]) u^(deg Total[#])/aut[deg, #] &, l[1]]];
g2[u_, v_, deg_] := Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &,l[nn]]];
Map[Reverse, Map[Select[#, # > 0 &] &,Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g1[u, v, 1] g2[u, 1, 1]^(q - 1) Product[g2[u, 1, d]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], {u,v}]]] // Flatten

Formula

T(n,k) = A002884(n)/A002884(n-k).

T(n,k) = T(n,k-1)*T(n-k+1,1).

Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = e(x)*g(y*x) where e(x) = Sum_{n>=0} x^n/B(n), g(x) = Sum_{n>=0} Product_{i=0..n-1} (q^n-q^i)*x^n/B(n), B(n) = Product_{i=0..n-1} (q^n-q^i)/(q-1)^n and q=2. - Geoffrey Critzer, Jan 03 2025

Extensions

Title improved by Geoffrey Critzer, Sep 16 2022