DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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The Lehmer code of a permutation

Definition

Let $σ = [σ (0), \dots, σ (n - 1)] \in S_{n}$ . Its Lehmer code is the tuple

$L (σ) = (ℓ_{1} (σ), \dots, ℓ_{n} (σ)) \in \prod_{i = 1}^{n} {0, \dots, i - 1},$

where

$ℓ_{i} (σ) : = ∣ {q \in n : n - i < q < n and σ (q) < σ (n - i)} ∣ .$

So $ℓ_{i} (σ)$ counts how many entries to the right of position $n - i$ are smaller than the entry in that position. This is the usual inversion table read from right to left, which is why the $i$ -th component ranges from $0$ to $i - 1$ .

For $n = 0$ , the Lehmer code is the unique element of the empty product.

Depends on

This page uses the published inversion number and the shifted Eulerian indexing $A(n,k)$

Used by

The Lehmer code is a bijection Sₙ→∏ᵢ₌₁ⁿ{0,…,i-1} Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Felix Gotti, MIT 18.211 Combinatorial Analysis, Lecture 5 (standard reference, not scraped)