Derangement

A permutation with no fixed points, so every element moves away from its original position.

Version

v1 · 2026-09-08 · History

Domain-specific #

4114

Origin domain

enumerative combinatorics

Subdomain

enumerative combinatorics

Aliases

Subfactorial permutation

Core Idea

The carrier is a permutation of a finite labeled set, not any disordering, partial derangements require separate definitions and subfactorial notation and nearest-integer formulas depend on integer n. Inclusion–exclusion subtracts permutations fixing at least one selected element from all n-factorial permutations, yielding n factorial times the truncated alternating exponential series. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

Derangement belongs to enumerative combinatorics and is useful where the analyst can specify the typed enumerative combinatorics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the finite set and size n, permutation, fixed point condition sigma(i)=i, requirement of zero fixed points, derangement number or subfactorial notation, inclusion–exclusion formula, recurrence and nearest n!/e relation, rencontres-number generalization and probability limit are explicit. The scope is broad within that domain but bounded by the need for the finite set and size n, permutation, fixed point condition sigma(i)=i, requirement of zero fixed points, derangement number or subfactorial notation, inclusion–exclusion formula, recurrence and nearest n!/e relation, rencontres-number generalization and probability limit are explicit.

Clarity

The abstraction clarifies a crowded vocabulary by making the finite set and size n, permutation, fixed point condition sigma(i)=i, requirement of zero fixed points, derangement number or subfactorial notation, inclusion–exclusion formula, recurrence and nearest n!/e relation, rencontres-number generalization and probability limit are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Derangement. Derangement compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed enumerative combinatorics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the finite set and size n, permutation, fixed point condition sigma(i)=i, requirement of zero fixed points, derangement number or subfactorial notation, inclusion–exclusion formula, recurrence and nearest n!/e relation, rencontres-number generalization and probability limit are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of enumerative combinatorics because they reuse the typed enumerative combinatorics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Inclusion–exclusion subtracts permutations fixing at least one selected element from all n-factorial permutations, yielding n factorial times the truncated alternating exponential series., and type the carrier, state every parameter and convention in the definition, test that the finite set and size n, permutation, fixed point condition sigma(i)=i, requirement of zero fixed points, derangement number or subfactorial notation, inclusion–exclusion formula, recurrence and nearest n!/e relation, rencontres-number generalization and probability limit are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Derangement Domain-specific

Parents (1) — more general patterns this builds on

• Derangement is a kind of Permutation Prime

  The proposed strict upward parent is prime:permutation.

Hierarchy paths (3) — routes to 1 parentless root

• Derangement → Permutation → Bijectivity → Function (Mapping)

Show alternative paths (2)

Neighborhood in Abstraction Space

Derangement sits in a crowded region of the domain-specific corpus (14^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Enumerative Combinatorics & Partitions (24 abstractions)

Nearest neighbors

• Lobb number — 0.93
• Schröder number — 0.93
• Eulerian number — 0.93
• Piecewise syndetic set — 0.92
• Motzkin number — 0.91

Computed from structural-signature embeddings · 2026-09-08