Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.6 |

Department of Mathematics and Industrial Engineering

Polytechnique Montréal and GERAD

Montréal, PQ H3T 1J4

Canada

Anaelle Hertz

Department of Physics

University of Toronto

Toronto, ON M5S 1A7

Canada

Hadrien Mélot

Computer Science Department-Algorithms Lab

University of Mons

7000 Mons

Belgium

**Abstract:**

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.)

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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