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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, 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
0,4
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
Cf. A053763 (column k=0), A002884 (main diagonal).
Sequence in context: A126150 A374370 A364509 * A291056 A248831 A227729
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Nov 21 2021
STATUS
approved