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.

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

Offset: 0

Author

Geoffrey Critzer, Dec 15 2017

Keywords

• nonn
• tabl

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.

Examples

			Triangle T(n,k) 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;
  ...
		

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.

Crossrefs

Cf. A132186 (row sums).

Programs

Maple

b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, b(n-1, k-1)+2^k*b(n-1, k)))
    end:
T:= (n,k)-> 2^(k*(n-k))*b(n, k):
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Dec 02 2024

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}]]]

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)).