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.

翻译：钟数数计分出一组美元元素的不同方法数, 图形钟数则计数一个图形的顶点组的非等值分区数为稳定的组数。 图形理论和整数序列之间的这种关系促使我们研究图中非等值颜色的平均颜色数的属性, 以发现新的非微小的钟数不平等。 示例用于说明我们的方法 。