OFFSET
1,2
COMMENTS
a(n) is the number of complete rooted binary trees with n leaves in which the internal nodes are labeled either white or black; the two children (subtrees) of a white node have no specified orientation, but the two children (subtrees) of a black node are labeled left and right. Thus the notion of isomorphism for these trees is partly planar (for the black nodes) and partly abstract (for the white nodes).
Finding a recurrence relation is an easy exercise. Finding an exact formula is probably very difficult or even impossible: compare the OEIS page for A001190 (Wedderburn-Etherington numbers).
LINKS
F. Bagherzadeh, M. R. Bremner, S. Madariaga, Jordan trialgebras and post-Jordan algebras, arXiv:1611.01214 [math.RA], 2016.
Murray Bremner, Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.
EXAMPLE
For n = 4 the 25 association types are as follows, where * is commutative and # is noncommutative; some assumptions have been made regarding the order of the factors for the commutative operation:
( ( X * X ) * X ) * X,
( ( X # X ) * X ) * X,
( ( X * X ) # X ) * X,
( ( X # X ) # X ) * X,
( X # ( X * X ) ) * X,
( X # ( X # X ) ) * X,
( X * X ) * ( X * X ),
( X * X ) * ( X # X ),
( X # X ) * ( X # X ),
( ( X * X ) * X ) # X,
( ( X # X ) * X ) # X,
( ( X * X ) # X ) # X,
( ( X # X ) # X ) # X,
( X # ( X * X ) ) # X,
( X # ( X # X ) ) # X,
( X * X ) # ( X * X ),
( X * X ) # ( X # X ),
( X # X ) # ( X * X ),
( X # X ) # ( X # X ),
X # ( ( X * X ) * X ),
X # ( ( X # X ) * X ),
X # ( ( X * X ) # X ),
X # ( ( X # X ) # X ),
X # ( X # ( X * X ) ),
X # ( X # ( X # X ) ).
MAPLE
BWT := table():
BWT[ 1 ] := 1:
for arity from 2 to 24 do
BWT[ arity ] := 0:
# commutative operation
for i to floor((arity-1)/2) do
BWT[ arity ] := BWT[ arity ] + ( BWT[arity-i] * BWT[i] )
od:
if arity mod 2 = 0 then
BWT[ arity ] := BWT[ arity ] + binomial( BWT[arity/2]+1, 2 )
fi:
# noncommutative operation
for i to arity-1 do
BWT[ arity ] := BWT[ arity ] + ( BWT[arity-i] * BWT[i] )
od
od:
seq(BWT[ n ], n=1..24);
MATHEMATICA
BWT[1] = 1; For[arity = 2, arity <= 24, arity++, BWT[arity] = 0; (* commutative operation *) For[i = 1, i <= Floor[(arity-1)/2], i++, BWT[arity] = BWT[arity] + (BWT[arity-i]*BWT[i])]; If[EvenQ[arity], BWT[arity] = BWT[arity] + Binomial[BWT[ arity/2]+1, 2]]; (* non commutative operation *) For[i = 1, i <= arity-1, i++, BWT[arity] = BWT[arity] + (BWT[arity-i]*BWT[i])]];
Table[BWT[n], {n, 1, 24}] (* Jean-François Alcover, Feb 15 2019, from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Murray R. Bremner, Aug 26 2016
STATUS
approved