Enumeration formulas involving partitions

[webonfim]

Neil,

In 2005, I searched this book, looking for those enumeration formulas involving partitions. Now after read the first pages of the chapters 1 and 4 of “Graphical Enumeration”, I continue to think that nobody published those formulas before.

In this book often they count connected graphs of a class from results on the total of graphs of that class. The formulas involving partitions can be used to solve problems found in the opposite direction, i.e., to count graphs of a class from results on the number of connected graphs of that class.

I think that I am over the shoulders of giants when I use those formulas because they depend on results on the number of connected graphs that are obtained with generating functions, Pólya’s theorem and group theory.

The “partition formulas” so far depending on the “real stuff “ can be used by anyone with elementary notions of combinatorics. If “the pros” had already found the numbers of connected graphs of a class, by a formula, or by an OEIS sequence, etc., we can use those numbers to count “several types” of  graphs with connected components of that class. Objects with a given number of components and/or with restrictions on the order of their components, are easily enumerated, both for labeled and unlabeled classes.

Washington

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