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 nontrivial inequalities for the Bell numbers. Examples are given to illustrate our approach.

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(Concerned with sequences A005493 A141390.)

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Received March 9 2021; revised versions received October 13 2021; December 6 2021. Published in Journal of Integer Sequences, December 21 2021.

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