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A combinatorial identity for rooted labeled forests.

Benjamin Hackl1

  • 1Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria.

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Summary
This summary is machine-generated.

This study provides a direct combinatorial proof linking rooted forests and set partitions. It also contextualizes this identity within forest volumes and multinomial identities.

Keywords:
Combinatorial identityForestHurwitz multinomial identitySet partition

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Area of Science:

  • Combinatorics
  • Discrete Mathematics

Background:

  • Rooted forests and unordered set partitions are fundamental combinatorial objects.
  • Identities connecting different combinatorial structures are crucial for advancing the field.

Purpose of the Study:

  • To present a straightforward combinatorial proof for an identity connecting rooted forests and unordered set partitions.
  • To provide context for this identity within the broader areas of forest volumes and multinomial identities.

Main Methods:

  • A direct combinatorial proof is employed.
  • The proof establishes a bijective or enumerative link between the two structures.

Main Results:

  • A novel combinatorial identity is proven, directly connecting rooted forests and unordered set partitions.
  • The identity is situated within the established literature on forest volumes and multinomial identities.

Conclusions:

  • The combinatorial proof offers a clear and accessible understanding of the relationship between rooted forests and set partitions.
  • This work contributes to the understanding of combinatorial identities and their applications.