Labelled objects

Labelled combinatorial classes have objects composed of \(N\) atoms, Labelled with the integers from 1 through \(N\).

Example 1. Urns

Def. An urn is a set of labelled atoms

Counting seq	EGF
\(U_N=1\)	\(e^z\)

Example 2. permutations

Def. A permutation is a sequence of labelled atoms.

Counting seq	EGF
\(P_N=N!\)	\(1/(1-z)\)

\[ \sum_{N \geq 0} \frac{N ! z^N}{N !}=\sum_{N \geq 0} z^N=\frac{1}{1-z} \]

Example 3. cycles

Def. A cycle is a cyclic sequence of labelled atoms

Counting seq	EGF
\(C_N=(N-1) !\)	\(\ln \frac{1}{1-z}\)

\[ \sum_{N \geq 1} \frac{(N-1) ! z^N}{N !}=\sum_{N \geq 1} \frac{z^N}{N}=\ln \frac{1}{1-z} \]