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

Using Graph Theory to Derive Inequalities for the Bell Numbers

Alain Hertz
Department of Mathematics and Industrial Engineering
Polytechnique Montréal and GERAD
Montréal, PQ H3T 1J4

Anaelle Hertz
Department of Physics
University of Toronto
Toronto, ON M5S 1A7

Hadrien Mélot
Computer Science Department-Algorithms Lab
University of Mons
7000 Mons


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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