Derangements

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Derangements are permutations with no rencontre (“rencontre” is french for “encounter”), i.e. no fixed-point.

Example

The 6 permutations of (1, 2, 3) are, in lexicographic order: {(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)} among which

• {(1, 2, 3)} (obviously) have 3 rencontres (fixed-points);
• ∅ (obviously) have 2 rencontres;
• {(1, 3, 2), (2, 1, 3), (3, 2, 1)} have 1 rencontre;
• {(2, 3, 1), (3, 1, 2)} have 0 rencontre.

The 2 derangements of (1, 2, 3) are thus {(2, 3, 1), (3, 1, 2)}.

See also

• Derangement numbers (number of permutations with no rencontre, i.e. no fixed-point)
• Number of permutations with k rencontres
• Arrangements