A382962 Number of symmetric ternary maps f : S X S X S -> S on a set S of n elements which can be represented as a superposition of binary maps * : S X S -> S.

1, 5, 48, 831, 21320, 772422

Offset

1,2

Comments

Sequence corresponds to Tcomm(n) from the Zusmanovich paper.

We consider ternary maps that can be represented as

f(x, y, z) = (x * y) * z.

Symmetry is defined here as independence from the order of arguments, hence

f(x, y, z) = f(x, z, y) = f(y, x, z) = f(y, z, x) = f(z, x, y) = f(z, y, x).

Question 15 from the Zusmanovich paper remains unanswered: at least for n <= 6, every f satisfying the above can be represented by a superposition of some commutative '*', but no proof that this is the case for all n.

Links

Table of n, a(n) for n=1..6.

Pasha Zusmanovich, Lie algebras and around: selected questions, arXiv:1608.05863 [math.RA], 2016.

Example

For N=2, 6 of the N^(N^2) binary maps '*' define a symmetric ternary map f(x,y,z)
  *:                  f:
  ((0, 0), (0, 0))    (((0, 0), (0, 0)), ((0, 0), (0, 0)))
  ((0, 0), (0, 1))    (((0, 0), (0, 0)), ((0, 0), (0, 1)))
  ((0, 1), (1, 0))    (((0, 1), (1, 0)), ((1, 0), (0, 1)))
  ((0, 1), (1, 1))    (((0, 1), (1, 1)), ((1, 1), (1, 1)))
  ((1, 0), (0, 1))    (((0, 1), (1, 0)), ((1, 0), (0, 1)))
  ((1, 1), (1, 1))    (((1, 1), (1, 1)), ((1, 1), (1, 1)))
The 3rd and 5th ternary maps are identical and are counted only once, hence a(2) = 5.

Crossrefs

Cf. A283840, A283841.

Sequence in context: A370758 A352254 A346183 * A224510 A333982 A063429

Adjacent sequences: A382959 A382960 A382961 * A382963 A382964 A382965

Keyword

nonn,hard,more

Author

Bert Dobbelaere, Apr 10 2025

Status

approved