Abstract
A very well known theorem of Cayley asserts that there are nn−2 distinct trees with vertex set {1,2, …, n}. We can express this result in the language of matroid theory by saying, “The complexity of the complete graph on n vertices is nn−2.”
Below we offer a proof of Cayley's Theorem that is based on Möbius Inversion in the generalized sense of Rota [2]. Proofs of Cayley's Theorem are legion, but the present one has the unusual feature that the inversion takes place over a poset that is not a lattice. The poset is an interesting combinatorial object that probably has other applications.
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References
Moon, J. Various proofs of Cayley's Formula for counting trees. Chapter 11 in A Seminar on Graph Theory, F. Harary, ed.
Rota, G.-C. On the foundations of combinatorial theory I : Theory of Mobius finctions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964) 340 – 368.
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Peele, R. (2010). The Poset of Subpartitions and Cayley's Formula for the Complexity of a Complete Graph. In: Barlotti, A. (eds) Matroid Theory and its Applications. C.I.M.E. Summer Schools, vol 83. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11110-5_5
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DOI: https://doi.org/10.1007/978-3-642-11110-5_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11109-9
Online ISBN: 978-3-642-11110-5
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