A342245 Number of ordered pairs (S,T) of n X n idempotent matrices over GF(2) such that ST = TS = S.

1, 3, 21, 339, 12483, 1074339, 219474243, 107174166147, 126918737362179, 367662330459585027, 2614066808849501254659, 45985259502347910886975491, 2009925824909891218929491103747, 218411680908756813835229484489351171, 59296916710446845619466630380450779971587

Offset: 0

Author

Geoffrey Critzer, Mar 07 2021

Keywords

• nonn

Comments

The components in the ordered pairs are not necessarily distinct.

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.

Crossrefs

Cf. A132186, A005329.

Programs

Mathematica

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

Formula

Let E(x) = Sum_{n>=0} x^n/(2^binomial(n,2) * [n]A005329(n).%20Then%20E(x)%5E3%20=%20Sum">2!) where [n]_2! = A005329(n). Then E(x)^3 = Sum{n>=0} a(n)x^n/(2^binomial(n,2) * [n]_2!)