Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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