Derangementsideks

Derangementsideks is a coined term used to describe a class of problems and narrative concepts in which a set of sidekick characters are reassigned to non-original roles so that no character remains in its canonical position. The term blends derangement, a permutation with no fixed points, with the idea of sidekicks.

In formal terms, for a set of n labeled sidekicks, a derangement is a permutation π of {1,...,n} with π(i) ≠ i for all i. The number of such derangements is !n, the subfactorial, given by !n = n! sum_{k=0}^n (-1)^k / k!, and approximately !n ≈ n!/e. Derangementsideks studies counting, existence, and construction of these reassigned roleings, as well as their narrative and puzzle-creation applications.

Examples: for n = 3, !3 = 2, corresponding to the permutations (2,3,1) and (3,1,2). These illustrate a complete reshuffling where no sidekick occupies their original position.

Applications include generating unpredictable team lineups in fiction, designing puzzles that require non-fixed role assignments, and algorithmic generation of non-fixed mappings under additional constraints (such as distance limits between original and new positions or prohibiting specific swaps).

Variants include partial derangements, where some elements may stay fixed, and constrained or layered derangements, which impose extra rules on permissible mappings. Derangementsideks thus links a classic combinatorial concept with creative and optimization contexts.

See also: Derangement; Subfactorial; Permutation; Combinatorics.

Notes: The term is informal and not a standard mathematical designation; it is primarily used in creative contexts or puzzle design to describe derangement-like role swaps.