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 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, 1, 3, 7, 22, 77, 314 (list; graph; refs; listen; history; text; internal format)
 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 semi-lattices, aka 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 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: A181769 A075214 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

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Last modified August 3 08:53 EDT 2020. Contains 336197 sequences. (Running on oeis4.)