Derangement Number

Count permutations of n labeled elements with no fixed points exactly, including one empty derangement; n is capped at 2,000.

Description

A derangement is a permutation with no fixed points.

A derangement is a permutation with no fixed points. Every labeled element must move to a different position, while the empty permutation counts as one.^[1]

The tool returns value. The indexing and output conventions below determine how to read these values.

When to use Derangement Number

Use this for the following task: Count permutations of n labeled elements with no fixed points exactly, including one empty derangement; n is capped at 2,000. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Derangement Number and understand its results

Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
Enter n according to the field table, then run the tool.
Read value. Preserve exact decimal strings and the stated indexing or coefficient order.
Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.

Input field	Meaning	Value type and bounds	Requirement
n	Permutation size (resource limit 2000)	integer; minimum 0; maximum 2000	Required

Use the two-term BigInt recurrence and return the exact count as a decimal string.

D(0)=1; D(1)=0; D(n)=(n−1)(D(n−1)+D(n−2))

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
value	Derangement count	string (integer/arbitrary)

json

{
  "input": {
    "n": 6
  },
  "output": {
    "value": "265"
  }
}

Limitations and assumptions

The set size is at most 2000. Permutations are counted rather than generated; use rencontres numbers when a nonzero exact fixed-point count is needed.

Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.

Alternative or Complementary methods

Choose Rencontres Number when you need its operation: Count permutations of n labeled elements with exactly k fixed points using C(n,k)·D(n−k), returning zero for k>n; indices are capped at 1,000. Choose Involution Number for the complementary operation: Count permutations of n labeled elements consisting only of fixed points and transpositions exactly, including the empty involution; n is capped at 2,000.

Background reference: OEIS A000166.

References

OEIS A000166 — OEIS Foundation

Similar or alternative tools

Rencontres Number
Count permutations of n labeled elements with exactly k fixed points using C(n,k)·D(n−k), returning zero for k>n; indices are capped at 1,000.
Involution Number
Count permutations of n labeled elements consisting only of fixed points and transpositions exactly, including the empty involution; n is capped at 2,000.