[seqfan] Need help in understanding a combstruct specification in sequence A052886.

[Antti Karttunen]

Dear SeqFans,

I wonder whether anybody with Maple there could elaborate what kind of
combinatorial structures the following combstruct grammar:
  spec := [S, {C=Set(Z, 1 <= card), S=Prod(B, C), B=Sequence(S)}, labeled]:
seq(combstruct[count](spec, size=n), n=0..20);
from https://oeis.org/A052886 produces?

A quick starter to the library is here:
https://www.maplesoft.com/support/help/Maple/view.aspx?path=combstruct%2fspecification

If somebody could just list for me all structures of size 3 (there should
be 19 of them), it would help a lot. I tried to do this by emulating
combstruct library manually, but as the structure is labeled, I soon lost
the track, and I'm not sure how PROD actually works (it's not commutative,
right? Judging from examples in A000108). I used to know this stuff better
myself, but the years have gone by since then.

BTW, there's a similar Haskell-library now:
https://hackage.haskell.org/package/species-0.2/docs/Math-Combinatorics-Species.html


Best regards (and Happy New Year!),

Antti