  The Bell numbers count the number of different ways to partition a set of $n$
elements while the graphical Bell numbers count the number of non-equivalent
partitions of the vertex set of a graph into stable sets. This relation between
graph theory and integer sequences has motivated us to study properties on the
average number of colors in the non-equivalent colorings of a graph to discover
new non trivial inequalities for the Bell numbers. Example are given to
illustrate our approach.
