1,3

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 semi-lattices, aka idempotent commutative semigroups, by finite Abelian groups with an odd number of elements and by hybrids of these two types.

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

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).

Cf. A111773 (total number).

Sequence in context: A148688 A075214 A070766 this_sequence A018190 A000228 A108070

Adjacent sequences: A111769 A111770 A111771 this_sequence A111773 A111774 A111775

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2005