A111772 Number of non-isomorphic Average systems with n elements. An Average system has one binary operation "avg" and satisfies the three axioms avg(A,A)=A, avg(A,B)=avg(B,A), avg(avg(A,B),avg(C,D)) = avg(avg(A,C),avg(B,D)).

1, 1, 3, 7, 22, 77, 314

Offset

1,3

Comments

Axiom 1 is idempotence; axiom 2 is commutativity. The only unfamiliar axiom is the third one, mid-quarter-swap, a kind of tree-editing axiom. Together with commutativity, it allows free permutation of nodes at each specific level of a binary tree representing an expression.

The Average axioms are also satisfied by lower semilattices, a.k.a., idempotent commutative semigroups, by finite Abelian groups with an odd number of elements and by hybrids of these two types.

References

Richard C. Schroeppel, Posting to Math-Fun Mailing List, May 01, 2005.

Links

Table of n, a(n) for n=1..7.

Example

Summary table:
n.Systems...Tables....Group orders
1.......1........1....1
2.......1........2....1
3.......3.......10....1 2 6
4.......7.......92....1.2 2.3 6.2
5......22.....1321....1.5 2.10 4 6.4 20 24
6......77....27882....1.19 2.31 4.7 6.12 12.4 20 24.2 120
7.....314...819330....1.85 2.122 4.32 6.36 8.4 12.19 20.2 24.6 36.2 42 48 72 120.2 720
n is the size of the system.
Systems is the count of non-isomorphic systems of that size.
Tables is the total number of tables, with no culling for isomorphism.
Group orders is the number of systems with each size of automorphism group.
For example, there are 314 non-isomorphic Average systems with 7 elements.
85 of those systems have the trivial automorphism group (only the identity),
and each system gives rise to 7! = 5040 distinct tables. There's one
system with an automorphism group of 720 elements, which gives rise to only
5040/720 = 7 different tables. The total number of possible 7-element tables
is 7^49, of which roughly 7^7 satisfy the Average rules.
We have the obvious identities 314 = 85 + 122 + 32 + ... + 1 + 2 + 1 and 819330 = 5040 * (85/1 + 122/2 + 32/4 + ... + 1/72 + 2/120 + 1/720).

Crossrefs

Cf. A111773 (total number).

Sequence in context: A075214 A360887 A070766 * A233005 A018190 A323930

Adjacent sequences: A111769 A111770 A111771 * A111773 A111774 A111775

Keyword

nonn,nice

Author

N. J. A. Sloane, Nov 21 2005

Status

approved