A349545 Triangular array read by rows: T(n,k) = A002884(k)*2^((n-k)(n-k-1)), n >= 0, 0 <= k <= n.

1, 1, 1, 4, 1, 6, 64, 4, 6, 168, 4096, 64, 24, 168, 20160, 1048576, 4096, 384, 672, 20160, 9999360, 1073741824, 1048576, 24576, 10752, 80640, 9999360, 20158709760, 4398046511104, 1073741824, 6291456, 688128, 1290240, 39997440, 20158709760, 163849992929280

Offset: 0

Author

Geoffrey Critzer, Nov 21 2021

Keywords

• nonn
• tabl

Comments

For A,B in the set of n X n matrices over GF(2) let A ~ B iff A^j = B^k for some positive j,k. Then ~ is an equivalence relation. There is exactly one idempotent matrix in each equivalence class. Let E be an idempotent matrix of rank k. Then T(n,k) is the size of the class containing E.

The classes in the equivalence relation described above are called the torsion classes corresponding to the idempotent E. - Geoffrey Critzer, Oct 02 2022

Examples

			Triangle begins:
        1;
        1,    1;
        4,    1,   6;
       64,    4,   6, 168;
     4096,   64,  24, 168, 20160;
  1048576, 4096, 384, 672, 20160, 9999360;
  ...
T(3,1)=4 because we have: { I = {{0, 0, 0}, {0, 0, 0}, {0, 0, 1}},
  A= {{0, 0, 0}, {1, 0, 0}, {0, 0, 1}}, B= {{0, 1, 0}, {0, 0, 0}, {0, 0, 1}},
  C= {{1, 1, 0}, {1, 1, 0}, {0, 0, 1}} } where I is idempotent of rank 1 and A^2=B^2=C^2=I.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• Encyclopedia of Mathematics, Periodic semigroup

Crossrefs

Cf. A053763 (column k=0), A002884 (main diagonal).

Programs

Mathematica

q = 2; nn = 7;Table[Table[Product[q^d - q^i, {i, 0, d - 1}] q^((n - d) (n - d - 1)), {d, 0,n}], {n, 0, nn}] // Grid

PARI

\\ here b(n) is A002884(n).
b(n) = {prod(i=2, n, 2^i-1)<Andrew Howroyd, Nov 22 2021