%I #30 Dec 19 2022 07:04:37

%S 1,1,2,6,21,78,308,1264,5332,22994,100896,449004

%N The number of plane rooted complete ternary trees with 2n+1 unlabeled leaves (hence n internal nodes including the root where n starts at 0) satisfying these two conditions: (1) if one of the three children of any internal node is the greatest in deglex order then that child is not the leftmost child; (2) if one of the three children of any internal node is the smallest in deglex order then that child is not the rightmost child. Deglex order refers to degree-lexicographical order defined inductively on the number of leaves (see details under Comments).

%C "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.

%C 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.

%C See further details in the comments to the Maple program which is attached as a a-file.

%H Murray R. Bremner, <a href="/A287211/a287211_2.txt">Association types up to and including arity 11</a>

%H Murray R. Bremner, <a href="/A287211/a287211_1.txt">Maple worksheet for generating association types</a>

%e 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.

%e Arity 1, number of types 1:

%e a.

%e Arity 3, number of types 1:

%e [abc].

%e Arity 5, number of types 2:

%e [ab[cde]],

%e [a[bcd]e].

%e Arity 7, number of types 6:

%e [ab[cd[efg]]],

%e [ab[c[def]g]],

%e [a[bcd][efg]],

%e [a[bc[def]]g],

%e [a[b[cde]f]g],

%e [[abc]d[efg]].

%p See attached a-file under Links.

%Y Cf. A357538, A000598, A000625, A001764, A006629, A036370, A036437, A233389.

%K nonn,more

%O 0,3

%A _Murray R. Bremner_, May 21 2017