OFFSET
0,3
COMMENTS
"Plane" means "embedded in the plane" or (equivalently) the three children of each internal node (including the root) are ordered left, middle, right. Deglex order on trees with 2n+1 leaves is defined as follows: to compare two such trees T and U with children T_1, T_2, T_3 and U_1, U_2, U_3, first find the least index 1 <= i <= 3 for T_i <> U_i, then compare T_i and U_i in deglex order already defined inductively on trees with fewer than 2n+1 leaves; note that this requires comparing trees with different numbers of leaves, so we say that T_i precedes U_i if either (i) T_i has fewer leaves than U_i, or (ii) T_i and U_i have the same number of leaves, and T_i precedes U_i in deglex order.
An alternative description of this sequence: it counts the distinct association types in arity 2n+1 for a ternary operation [a,b,c] satisfying the cyclic-sum relation [a,b,c] + [b,c,a] + [c,a,b] = 0. The two conditions stated under "Name" are necessary to deal with the possibility of repeated factors: [a,a,b], [a,b,a], [b,a,a] where a < b in deglex order, and [a,b,b], [b,a,b], [b,b,a] where a < b in deglex order.
See further details in the comments to the Maple program which is attached as a a-file.
LINKS
Murray R. Bremner, Association types up to and including arity 11
Murray R. Bremner, Maple worksheet for generating association types
EXAMPLE
Association types for arities 1, 3, 5, 7 are as follows in deglex order. See Links for a-file with association types for arities up to 11.
Arity 1, number of types 1:
a.
Arity 3, number of types 1:
[abc].
Arity 5, number of types 2:
[ab[cde]],
[a[bcd]e].
Arity 7, number of types 6:
[ab[cd[efg]]],
[ab[c[def]g]],
[a[bcd][efg]],
[a[bc[def]]g],
[a[b[cde]f]g],
[[abc]d[efg]].
MAPLE
See attached a-file under Links.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Murray R. Bremner, May 21 2017
STATUS
approved