Permutation calculator

Definition

Let S be an n-element set (n ∈ ℕ₀). A permutation of S is a bijection π: S → S. Equivalently, when the elements of S are written as a sequence (x₁, …, xₙ) with no repeats, each such sequence is one permutation. The number of all permutations is |Sym(S)| = n!, where n! = 1·2·…·n for n ≥ 1 and 0! = 1 by the empty-product convention. In school notation and on this calculator we write P(n) = n!. The identity P(n,k) = n!/(n−k)! for 0 ≤ k ≤ n counts k-permutations (variations without repetition); when k = n we get P(n,n) = n!/0! = n!, so P(n) = P(n,n).

The definition is standard enumerative combinatorics: Brualdi, Introductory Combinatorics; Stanley, Enumerative Combinatorics, vol. 1; Graham, Knuth, Patashnik, Concrete Mathematics (factorial and counting chapters). MathWorld (Permutation) and Wikipedia (Permutation) state the same equality |Sₙ| = n! for the symmetric group. Computation here: n = ⌊a⌋; result = product 2…n when n ≥ 0, otherwise 0.

The calculator does not list permutations and does not classify cycles, transpositions, or the sign of a permutation (sgn). The result is only the cardinality of all linear arrangements — the integer P(n).

Historical and conceptual context

Counting arrangements appears in ancient Indian texts and in medieval writings on “orders of ranks.” Modern formalization ties permutations to the symmetric group Sₙ (Galois, Cauchy) and to the factorial as a counting function. Christian Kramp (1808) helped popularize the n! notation; the link n! = Γ(n+1) for n ∈ ℕ₀ connects combinatorics to the gamma function (NIST DLMF §5.2). In classical probability each of the n! permutations of n distinct outcomes is often treated as equally likely — hence n! in denominators of classical models (for example, lining up n people).

Conceptually separate: (1) a permutation as a bijection / linear line-up — this page; (2) permutations with repetition n!/(n₁!…nᵣ!) when copies are indistinguishable; (3) k-permutations P(n,k); (4) combinations C(n,k). This calculator answers only (1) with n = ⌊input⌋ ≥ 0. The neighbouring factorial page shares the same product but treats negative inputs differently.

In classical cryptography the number of substitutions of an n-letter alphabet is n! — the order of the key space for a simple substitution cipher. In computer science n! appears in the cost of generating all orders and in sorting lower bounds (worst-case comparisons ≥ log₂(n!)).

Formal properties

Mathematical domain: n ∈ ℕ₀. Codomain: {1, 2, 6, 24, 120, …} with P(0) = P(1) = 1. Recurrence: P(0) = 1, P(n) = n·P(n−1) for n ≥ 1. Monotonicity: for n ≥ 1, P(n) ≥ P(n−1), strict when n ≥ 2.

Identities: P(n) = V(n,n) = n!/(n−n)!; C(n,k)·k! = P(n,k); in particular C(n,n)·n! = n! = P(n). Stirling’s approximation n! ∼ √(2πn)·(n/e)ⁿ estimates magnitude for large n (Concrete Mathematics; Feller). The group Sₙ has order n! and is generated by adjacent transpositions.

In this calculator the practical domain also allows non-integer inputs (floor) and replaces negative n with 0. That is a convention of this calculator, not a theorem about the Γ function: factorial is undefined for negative integers in ℕ, and Γ has poles at non-positive integers.

Full derivation

Step 1 (multiplication rule). Place n distinguishable objects in n seats. Seat 1: n choices. Seat 2: n−1 remaining. … Seat n: 1 choice. Total n·(n−1)·…·1.

Step 2 (factorial definition). Define n! := n·(n−1)·…·1 for n ≥ 1 and 0! := 1. Hence P(n) = n!.

Step 3 (recurrence). For n ≥ 1 factor out n: n! = n·(n−1)!, so P(n) = n·P(n−1). The base P(0) = 1 closes the induction.

Step 4 (link to P(n,k)). The number of injections from a k-set into an n-set is P(n,k) = n!/(n−k)!. When k = n an injection is a bijection, the denominator (n−n)! = 0! = 1, so P(n,n) = n!.

Step 5 (numeric check). For n = 5: 5·4·3·2·1 = 120. For n = 4: 24. For n = 0: 1. These match the browser loop r*=i.

Special cases

P(0) = 1 and P(1) = 1 — same number, different reading (empty arrangement vs one person). P(2) = 2: objects A,B give AB and BA. P(3) = 6. P(6) = 720. P(7) = 5040. P(10) = 3,628,800.

Input 5.7 → ⌊5.7⌋ = 5 → 120. Input −1 → 0 on this page. Permutations with repetition (for example letters in a word with repeats) require dividing by multiplicity factorials — this calculator does not. A uniform random permutation on Sₙ assigns probability 1/n! to each line-up.

When n is very large, the floating-point result first loses integer bit exactness (beyond 2⁵³), and above 170! it would exceed the floating-point range — for n ≥ 171 the calculator reports an error. That is a floating-point arithmetic limit, not a defect in the definition of n!.

Numerical notes

Computation: n = ⌊a⌋; P(n) = 1·2·…·n for n ≥ 0, otherwise 0. For n = 0 and n = 1 the loop does not multiply and r stays 1. Successive multiplication is exact for integer n! while the value stays within the exactly representable integers, that is through 18!. From 19! the panel shows the result as an approximation ≈ m × 10^e, and n ≥ 171 gives an error message.

The calculator accepts a comma or a period as the decimal separator. The calculator does not use big-integer arithmetic or arbitrary-precision libraries — an educational floating-point arithmetic result.

Neighbour comparison: the factorial page often returns 1 for negatives (via max(0,⌊x⌋)); here a negative returns 0 so the calculator does not pretend a “negative arrangement count” equals 0!.

Related notions and sources

Related on this site: factorial calculator (same product), variations V(n,k), combinations C(n,k). Related in theory: the group Sₙ, permutation sign, cycle type, Stirling numbers of the first kind (cycles), Stirling’s approximation for n!.

Sources: Brualdi — Introductory Combinatorics; Stanley — Enumerative Combinatorics vol. 1; Graham, Knuth, Patashnik — Concrete Mathematics; Feller — An Introduction to Probability Theory and Its Applications (n! in classical models); NIST DLMF §5.2 (factorial / gamma); MathWorld — Permutation; Wikipedia — Permutation / Factorial. Educational tool.