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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, 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

Table of n, a(n) for n=0..44.

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 A265603 A174186

Adjacent sequences:  A296545 A296546 A296547 * A296549 A296550 A296551

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Dec 15 2017

STATUS

approved

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Last modified July 19 04:25 EDT 2019. Contains 325144 sequences. (Running on oeis4.)