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

Offset

0,2

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.

Links

Table of n, a(n) for n=0..13.

Formula

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!)

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]

Crossrefs

Cf. A132186, A005329.

Sequence in context: A125054 A113085 A240936 * A332928 A083228 A052445

Adjacent sequences: A342242 A342243 A342244 * A342246 A342247 A342248

Keyword

nonn

Author

Geoffrey Critzer, Mar 07 2021

Status

approved