A286331 Triangle read by rows: T(n,k) is the number of n X n matrices of rank k over F_2.
1, 1, 1, 1, 9, 6, 1, 49, 294, 168, 1, 225, 7350, 37800, 20160, 1, 961, 144150, 4036200, 19373760, 9999360, 1, 3969, 2542806, 326932200, 8543828160, 39687459840, 20158709760, 1, 16129, 42677334, 23435953128, 2812314375360, 71124337751040, 325139829719040, 163849992929280
Offset: 0
Examples
Triangle T(n,k) begins:
1;
1, 1;
1, 9, 6;
1, 49, 294, 168;
1, 225, 7350, 37800, 20160;
1, 961, 144150, 4036200, 19373760, 9999360;
...
T(2,1) = 9 because there are 9, 2 X 2 matrices in F_2 that have rank 1: {{0, 0}, {0, 1}}, {{0, 0}, {1, 0}}, {{0, 0}, {1, 1}}, {{0, 1}, {0, 0}}, {{0, 1}, {0, 1}}, {{1, 0}, {0, 0}}, {{1, 0}, {1, 0}}, {{1,1}, {0, 0}}, {{1, 1}, {1, 1}}.
Links
- Robert Israel, Table of n, a(n) for n = 0..1710 (rows 0 to 57, flattened)
- Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
- Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
- Wikipedia, q-binomial
Programs
-
Maple
T:= (n,k) -> mul((2^n-2^j)^2/(2^k-2^j),j=0..k-1): seq(seq(T(n,k),k=0..n),n=0..10); # Robert Israel, May 15 2017
-
Mathematica
q = 2; Table[Table[Product[(q^n - q^i)^2/(q^k - q^i), {i, 0, k - 1}], {k, 0, n}], {n, 0, 6}] // Grid
Formula
T(n,k) = Product_{j=0..k-1} (2^n - 2^j)^2/(2^k - 2^j) = A022166(n,k) * Product_{j=0..k-1} (2^n - 2^j).