Digraphs with exactly one Eulerian tour
We give two combinatorial proofs of the fact that the number of loopless digraphs on the vertex set $[n]$ with no isolated vertices and with exactly one Eulerian tour up to a cyclic shift is $\frac{1}{2}(n-1)!C_{n}$, where $C_{n}$ denotes the $n$-th Catalan number. We construct a bijection with a se...
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| Sprache: | eng |
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| Zusammenfassung: | We give two combinatorial proofs of the fact that the number of loopless
digraphs on the vertex set $[n]$ with no isolated vertices and with exactly one
Eulerian tour up to a cyclic shift is $\frac{1}{2}(n-1)!C_{n}$, where $C_{n}$
denotes the $n$-th Catalan number. We construct a bijection with a set of
labeled rooted plane trees and with a set of valid parenthesis arrangements. |
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| DOI: | 10.48550/arxiv.2104.10734 |