• Corpus ID: 232478457

Using Graph Theory to Derive Inequalities for the Bell Numbers

@article{Hertz2021UsingGT,
  title={Using Graph Theory to Derive Inequalities for the Bell Numbers},
  author={Alain Hertz and Anaelle Hertz and Hadrien M'elot},
  journal={ArXiv},
  year={2021},
  volume={abs/2104.00552}
}
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. 

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References

SHOWING 1-10 OF 12 REFERENCES
On the Average Rank of an Element in a Filter of the Partition Lattice
  • K. Engel
  • Mathematics, Computer Science
    J. Comb. Theory, Ser. A
  • 1994
TLDR
It is proved that (1 |F| ) Σ πϵF r(π) ≧ ( 1 |P n | ) Φ p(A, λ + 1)/(λ + 1) n holds for all positive integers λ.
On the Number of Distinct Block Sizes in Partitions of a Set
TLDR
The average number of distinct block sizes in a partition of a set of n elements is asymptotic to e log n as n → ∞, which is in striking contrast to the fact that the average total number of blocks in a partitions is ∼ n (log n ) −1 as n→ ∞.
A sharp lower bound on the number of non-equivalent colorings of graphs of order and maximum degree
TLDR
A sharp lower bound on B ( G ) is determined for graphs G of order n and maximum degree n − 3 , and the graphs for which the bound is attained are characterized.
Counting the number of non-equivalent vertex colorings of a graph
TLDR
Borders on B ( G ) for graphs with a maximum degree constraint are studied and all graphs that reach the bound with equality are described.
Stirling numbers of the second kind and Bell numbers for graphs
TLDR
This paper summarizes the known properties of Stirling numbers of the second kind and Bell numbers for graphs, and proves new results about them, which give an alternative way to study r-StirlingNumbers of the first kind and r-Bell numbers.
Stirling Numbers of Forests and Cycles
TLDR
This note considers Stirling numbers of forests, and establishes asymptotic normality for the number of classes in a uniformly chosen partition of $C_n$ (the cycle on $n$ vertices) into non-empty independent sets.
Bell and Stirling numbers for disjoint unions of graphs
  • Congressus Numerantium
  • 2010
Bell and Stirling Numbers for Graphs
The Bell number B(G) of a simple graph G is the number of partitions of its vertex set whose blocks are independent sets of G. The number of these partitions with k blocks is the (graphical) Stirling
Chromatic polynomials and chro-maticity of graphs
# The Number of -Colourings and Its Enumerations # Chromatic Polynomials # Chromatic Equivalence of Graphs # Chromaticity of Multi-Partite Graphs # Chromaticity of Subdivisions of Graphs # Graphs in
Engel's Inequality for Bell Numbers
  • E. R. Canfield
  • Mathematics, Computer Science
    J. Comb. Theory, Ser. A
  • 1995
TLDR
It is proved that Engel's conjecture for all n sufficiently large by an extension of the Moser-Wyman asymptotic formula for the Bell numbers is proved.
...
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