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
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
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Dec 15 2017
STATUS
approved