OFFSET
0,4
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
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Encyclopedia of Mathematics, Periodic semigroup
EXAMPLE
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.
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
PROG
(PARI) \\ here b(n) is A002884(n).
b(n) = {prod(i=2, n, 2^i-1)<<binomial(n, 2)}
T(n, k) = {b(k)*2^((n-k)*(n-k-1))} \\ Andrew Howroyd, Nov 22 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Nov 21 2021
STATUS
approved