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.
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
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Mar 07 2021
STATUS
approved