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A173990 Dimensions of components for the operad of level algebras 0
1, 1, 3, 13, 75, 530, 4449, 43236 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) is the dimension of the space of multilinear operations with n arguments built from one commutative nonassociative binary operation * that satisfies the identity (a*b)*(c*d)=(a*c)*(b*d).

LINKS

Table of n, a(n) for n=1..8.

D. Chataur and M. Livernet, Adem-Cartan operads, arXiv:math/0209363 [math.AT], 2002-2003; Communications in Algebra 33 (2005), 4337-4360.

EXAMPLE

For n=4 there are 15 operations on four arguments x,y,z,t built from a general commutative nonassociative binary operation (see A001147). In our case, there are two relations between them: (x*y)*(z*t)=(x*z)*(y*t) and (x*t)*(y*z)=(x*z)*(y*t), so the dimension is 15-2=13.

PROG

(Haskell)

:m Math.Operad

let a = corolla 1 [1, 2]

let t1_23 = shuffleCompose 2 [1, 2, 3] a a

let t12_34 = shuffleCompose 1 [1, 2, 3, 4] t1_23 a

let t13_24 = shuffleCompose 1 [1, 3, 2, 4] t1_23 a

let t14_23 = shuffleCompose 1 [1, 4, 2, 3] t1_23 a

let r1 = (oet t12_34) - (oet t13_24) :: OperadElement Integer Rational PathPerm

let r2 = (oet t14_23) - (oet t13_24) :: OperadElement Integer Rational PathPerm

let gens = [r1, r2]

let cors = [a]

let gb = operadicBuchberger gens

let lms = map leadingMonomial gb

let computeStep d = basisElements cors lms d

map (length . computeStep) [1..]

CROSSREFS

Sequence in context: A330047 A334637 A007178 * A276924 A343788 A276895

Adjacent sequences:  A173987 A173988 A173989 * A173991 A173992 A173993

KEYWORD

nonn,more

AUTHOR

Vladimir Dotsenko, Mar 04 2010

STATUS

approved

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Last modified January 21 05:13 EST 2022. Contains 350473 sequences. (Running on oeis4.)