0,1

A sweet Boolean function is a monotone function whose BDD (binary decision diagram) is the same as the ZDD (zero-suppressed decision diagram) for its prime implicants (aka minimal solutions).

Equivalently, this is the number of sweet antichains contained in {1,...,n}. (Also called sweet clutters.) A sweet antichain whose largest element is n is a family of subsets A \cup (n\cup B) where A and B are sweet antichains in {1,...n-1}, B is nonempty and every element of A properly contains some element of B.

The property of being "sweet" depends on the order of the variables - compare A114491.

D. E. Knuth, The Art of Computer Programming, Section 7.1.4, in preparation.

All six of the antichains in {1,2} are sweet. They are emptyset, {emptyset}, {{1}}, {{2}}, {{1,2}} and {{1},{2}}.

Only 18 of the 20 antichains in {1,2,3} are sweet. The nonsweet ones are {{1,3},{2}} and {{1},{2,3}}. Because, in the latter case, A={1} and B={2}. However, {{1,2},{3}} is sweet because A={{1,2}} and B={emptyset}.

Some of the most interesting members of this apparently new family of Boolean functions are the connectedness functions, defined on the edges of any graph. The function f=[these arcs give a connected subgraph] is sweet, under any ordering of the arcs. Threshold functions [x_1+...+x_n \ge k] are sweet too.

Also the conjunction of sweet functions on disjoint sets of variables is sweet.

Cf. A114303, A114492, A114572.

Sequence in context: A038056 A072241 A093468 this_sequence A000304 A000614 A018290

Adjacent sequences: A114299 A114300 A114301 this_sequence A114303 A114304 A114305

nonn

D. E. Knuth, Aug 16 2008