Counting sets with exceptions

Counting numerical sets with no small atoms

A numerical set S with Frobenius number g is a set of integers with min(S) = 0 and max(Z − S) = g, and its atom monoid is A(S) = {n ∈ Z | n+ s ∈ S for all s ∈ S}. Let γg be the number of numerical sets S having A(S) = {0} ∪ (g,∞) divided by the total number of numerical sets with Frobenius number g. We show that the sequence {γg} is decreasing and converges to a number γ∞ ≈ .4844 (with accuracy...

Primitive Sets with Large Counting Functions

A set of positive integers is said to be primitive if no element of the set is a multiple of another. If S is a primitive set and S(x) is the number of elements of S not exceeding x, then a result of Erdős implies that ∫∞ 2 (S(t)/t log t) dt converges. We establish an approximate converse to this theorem, showing that if F satisfies some mild conditions and ∫∞ 2 (F (t)/t log t) dt converges, th...

Counting Lattice Paths with Various Step Sets

Finite Sets and Counting

These notes present an approach to obtaining the basic properties of the natural numbers in terms of the properties of finite sets (which are introduced independently of the natural numbers). The role played by the usual recursion theorem is taken over by a construction which can be regarded as a recursion theorem for finite sets.

Notes on counting finite sets