OFFSET
0,2
COMMENTS
Proof of formula by Robert Israel: If f is monotone, then for each x in X the set G(x) = {A in 2^X: x in f(A)} is an upset, i.e. if A is in G(x) and A \subset B then B is in G(x). Conversely, if for each x in X the set G(x) is an upset, then f is monotone. And the family {G(x): x in X} determines f, since f(A) = {x: A is in G(x)}. So the cardinality of the set of monotone set-functions is N(|X|)^|X| where N(|X|) is the cardinality of the set of upsets G of 2^X, or equivalently monotone Boolean functions. That is sequence A000372.
This sequence was motivated by a question by Federico Echenique on sci.math.research.
FORMULA
a(n) = A000372(n)^n.
CROSSREFS
KEYWORD
nonn
AUTHOR
W. Edwin Clark, Nov 06 2003
STATUS
approved