A276277 Association types for monomials with n arguments in an algebra with two binary operations, one commutative, one noncommutative.

1, 2, 6, 25, 111, 540, 2736, 14396, 77649, 427608, 2392866, 13570386, 77815161, 450418536, 2628225684, 15443406868, 91301938365, 542704450806, 3241411991712, 19443499011192, 117084197728737, 707532791560272, 4289252607915012, 26078561954153631

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

Table of n, a(n) for n=1..24.

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

Cf. A001190, A000108, A226909, A268172.

Sequence in context: A003454 A199241 A366236 * A355862 A299098 A229042

Adjacent sequences: A276274 A276275 A276276 * A276278 A276279 A276280

Keyword

nonn

Author

Murray R. Bremner, Aug 26 2016

Status

approved