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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.

%I A296548 #28 Dec 02 2024 17:13:53
%S A296548 1,1,1,1,6,1,1,28,28,1,1,120,560,120,1,1,496,9920,9920,496,1,1,2016,
%T A296548 166656,714240,166656,2016,1,1,8128,2731008,48377856,48377856,2731008,
%U A296548 8128,1,1,32640,44216320,3183575040,13158776832,3183575040,44216320,32640,1
%N 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.
%C A296548 Equivalently, T(n,k) is the number of n X n matrices, P, over GF(2) with rank k, such that P^2 = P.
%C A296548 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.
%H A296548 Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.
%H A296548 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%F A296548 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).
%F A296548 T(n,k) = A002884(n)/(A002884(k)*A002884(n-k)) = A022166(n,k)*2^(k(n-k)).
%e A296548 Triangle T(n,k) begins:
%e A296548   1;
%e A296548   1,    1;
%e A296548   1,    6,      1;
%e A296548   1,   28,     28,      1;
%e A296548   1,  120,    560,    120,      1;
%e A296548   1,  496,   9920,   9920,    496,    1;
%e A296548   1, 2016, 166656, 714240, 166656, 2016, 1;
%e A296548   ...
%p A296548 b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A296548       `if`(n=0, 1, b(n-1, k-1)+2^k*b(n-1, k)))
%p A296548     end:
%p A296548 T:= (n,k)-> 2^(k*(n-k))*b(n, k):
%p A296548 seq(seq(T(n, k), k=0..n), n=0..8);  # _Alois P. Heinz_, Dec 02 2024
%t A296548 nn = 8; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[
%t A296548     QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,
%t A296548   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}]]]
%Y A296548 Cf. A132186 (row sums).
%K A296548 nonn,tabl
%O A296548 0,5
%A A296548 _Geoffrey Critzer_, Dec 15 2017