A111772 - OEIS

%I #8 Sep 16 2024 12:00:34

%S 1,1,3,7,22,77,314

%N 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)).

%C 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.

%C 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.

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

%e Summary table:

%e n.Systems...Tables....Group orders

%e 1.......1........1....1

%e 2.......1........2....1

%e 3.......3.......10....1 2 6

%e 4.......7.......92....1.2 2.3 6.2

%e 5......22.....1321....1.5 2.10 4 6.4 20 24

%e 6......77....27882....1.19 2.31 4.7 6.12 12.4 20 24.2 120

%e 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

%e n is the size of the system.

%e Systems is the count of non-isomorphic systems of that size.

%e Tables is the total number of tables, with no culling for isomorphism.

%e Group orders is the number of systems with each size of automorphism group.

%e For example, there are 314 non-isomorphic Average systems with 7 elements.

%e 85 of those systems have the trivial automorphism group (only the identity),

%e and each system gives rise to 7! = 5040 distinct tables. There's one

%e system with an automorphism group of 720 elements, which gives rise to only

%e 5040/720 = 7 different tables. The total number of possible 7-element tables

%e is 7^49, of which roughly 7^7 satisfy the Average rules.

%e 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).

%Y Cf. A111773 (total number).

%K nonn,nice

%O 1,3

%A _N. J. A. Sloane_, Nov 21 2005