A349545 - OEIS

%I #27 Jan 07 2024 17:17:22

%S 1,1,1,4,1,6,64,4,6,168,4096,64,24,168,20160,1048576,4096,384,672,

%T 20160,9999360,1073741824,1048576,24576,10752,80640,9999360,

%U 20158709760,4398046511104,1073741824,6291456,688128,1290240,39997440,20158709760,163849992929280

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

%C 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.

%C The classes in the equivalence relation described above are called the torsion classes corresponding to the idempotent E. - _Geoffrey Critzer_, Oct 02 2022

%H Andrew Howroyd, <a href="/A349545/b349545.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%H Encyclopedia of Mathematics, <a href="http://encyclopediaofmath.org/index.php?title=Periodic_semi-group">Periodic semigroup</a>

%e Triangle begins:

%e         1;

%e         1,    1;

%e         4,    1,   6;

%e        64,    4,   6, 168;

%e      4096,   64,  24, 168, 20160;

%e   1048576, 4096, 384, 672, 20160, 9999360;

%e   ...

%e T(3,1)=4 because we have: { I = {{0, 0, 0}, {0, 0, 0}, {0, 0, 1}},

%e   A= {{0, 0, 0}, {1, 0, 0}, {0, 0, 1}}, B= {{0, 1, 0}, {0, 0, 0}, {0, 0, 1}},

%e   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.

%t 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

%o (PARI) \\ here b(n) is A002884(n).

%o b(n) = {prod(i=2, n, 2^i-1)<<binomial(n, 2)}

%o T(n,k) = {b(k)*2^((n-k)*(n-k-1))} \\ _Andrew Howroyd_, Nov 22 2021

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

%K nonn,tabl

%O 0,4

%A _Geoffrey Critzer_, Nov 21 2021