A114302 Number of "sweet" Boolean functions of n variables.

2, 3, 6, 18, 106, 2102, 456774, 7108935325

Offset: 0

Author

Don Knuth, Aug 16 2008

Keywords

• nonn
• more

Comments

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.

Examples

			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 >= k] are sweet too.
Also the conjunction of sweet functions on disjoint sets of variables is sweet.
		

References

• Donald E. Knuth, The Art of Computer Programming, Vol. 4, fascicle 1, section 7.1.4, p. 117, Addison-Wesley, 2009.

Crossrefs

Cf. A114303, A114492, A114572.