Geodesic
The set of ratios of derangements to permutations in digraphs is dense in \([0,1/2]\)
The electronic journal of combinatorics, Tome 29 (2022) no. 1
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A permutation in a digraph $G=(V, E)$ is a bijection $f:V \rightarrow V$ such that for all $v \in V$ we either have that $f$ fixes $v$ or $(v, f(v)) \in E$. A derangement in $G$ is a permutation that does not fix any vertex. Bucic, Devlin, Hendon, Horne and Lund proved that in any digraph, the ratio of derangements to permutations is at most $1/2$. Answering a question posed by Bucic, Devlin, Hendon, Horne and Lund, we show that the set of possible ratios of derangements to permutations in digraphs is dense in the interval $[0, 1/2]$.
DOI : 10.37236/10293
Classification : 05C20, 05C30, 05A05
Mots-clés : ratios of derangements, number of permutations 
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     author = {Bethany Austhof and Patrick Bennett and Nick Christo},
     title = {The set of ratios of derangements to permutations in digraphs is dense in \([0,1/2]\)},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {1},
     doi = {10.37236/10293},
     zbl = {1481.05061},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10293/}
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Bethany Austhof; Patrick Bennett; Nick Christo. The set of ratios of derangements to permutations in digraphs is dense in \([0,1/2]\). The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/10293

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