%I #8 Mar 07 2021 18:00:08

%S 1,3,21,339,12483,1074339,219474243,107174166147,126918737362179,

%T 367662330459585027,2614066808849501254659,45985259502347910886975491,

%U 2009925824909891218929491103747,218411680908756813835229484489351171

%N Number of ordered pairs (S,T) of n X n idempotent matrices over GF(2) such that ST = TS = S.

%C The components in the ordered pairs are not necessarily distinct.

%C The relation S<=T iff ST=TS=S gives a partial ordering on the idempotent matrices enumerated in A132186. Each length k chain (from bottom to top) in the poset corresponds to an ordered direct sum decomposition of GF(2)^n into exactly k subspaces.

%F Let E(x) = Sum_{n>=0} x^n/(2^binomial(n,2) * [n]_2!) where [n]_2! = A005329(n). Then E(x)^3 = Sum_{n>=0} a(n)x^n/(2^binomial(n,2) * [n]_2!)

%t nn = 13; b[n_] := q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 2;

%t e[x_] := Sum[x^n/b[n], {n, 0,nn}];Table[b[n],n,0,nn}]CoefficientList[Series[e[x]^3, {x, 0, nn}], x]

%Y Cf. A132186, A005329.

%K nonn

%O 0,2

%A _Geoffrey Critzer_, Mar 07 2021