login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A296548 Triangle read by rows: T(n,k) is the number of diagonalizable n X n matrices over GF(2) that have rank k, n >= 0, 0 <= k <= n. 4
1, 1, 1, 1, 6, 1, 1, 28, 28, 1, 1, 120, 560, 120, 1, 1, 496, 9920, 9920, 496, 1, 1, 2016, 166656, 714240, 166656, 2016, 1, 1, 8128, 2731008, 48377856, 48377856, 2731008, 8128, 1, 1, 32640, 44216320, 3183575040, 13158776832, 3183575040, 44216320, 32640, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Equivalently, T(n,k) is the number of n X n matrices, P, over GF(2) with rank k, such that P^2 = P.
Equivalently, T(n,k) is the number of direct sum decompositions of the vector space GF(2)^n into exactly two subspaces U and W such that the dimension of U is k.
LINKS
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.
FORMULA
T(n,k)/A002884(n) is the coefficient of y^k*x^n in the expansion of Sum_{n>=0} x^n\A002884(n) * Sum_{n>=0} y*x^n\A002884(n).
T(n,k) = A002884(n)/(A002884(k)*A002884(n-k)) = A022166(n,k)*2^(k(n-k)).
EXAMPLE
Triangle begins:
1;
1, 1;
1, 6, 1;
1, 28, 28, 1;
1, 120, 560, 120, 1;
1, 496, 9920, 9920, 496, 1;
1, 2016, 166656, 714240, 166656, 2016, 1;
MATHEMATICA
nn = 8; g[n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,
Table[g[n], {n, 0, nn}] CoefficientList[Series[Sum[(u z)^r/g[r] , {r, 0, nn}] Sum[z^r/g[r], {r, 0, nn}], {z, 0, nn}], {z, u}]]]
CROSSREFS
Cf. A132186 (row sums).
Sequence in context: A166960 A155908 A105373 * A201461 A340475 A368848
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Dec 15 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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 21:57 EDT 2024. Contains 371254 sequences. (Running on oeis4.)