login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

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).

EXAMPLE

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}}.

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

CROSSREFS

Main diagonal is A002884.

Column for k = 1 is A060867.

Row sums are A002416.

Sequence in context: A199082 A220669 A064230 * A089479 A269444 A199431

Adjacent sequences:  A286328 A286329 A286330 * A286332 A286333 A286334

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, May 07 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 20 08:16 EST 2018. Contains 317385 sequences. (Running on oeis4.)