Enumerating labeled graphs with a k-cycle

TL;DR

This paper initially aimed to enumerate labeled graphs containing a specific cycle length but was withdrawn due to the existence of a more general known result in combinatorial enumeration.

Contribution

The paper's specific contribution was the enumeration of labeled graphs with a k-cycle, but it was withdrawn after recognizing a more comprehensive existing result.

Findings

Enumeration of labeled graphs with a k-cycle was initially proposed.

The result was superseded by a more general theorem in combinatorial enumeration.

The paper was withdrawn due to referencing an existing comprehensive result.

Abstract

This paper has been withdrawn by the author as a more general result is known and could be referenced at "Combinatorial Enumeration" by Ian P. Goulden, David M. Jackson - Dover Publication 2004 - Section 3.3.13, Page 176.