Permutation Lexicographic Rank

Return a zero-based exact lexicographic permutation rank as decimal integer text; no floating factorial approximation.

Description

Return a zero-based exact lexicographic permutation rank as decimal integer text; no floating factorial approximation.

A Lehmer code counts smaller values remaining to the right. Factorial weights turn those digits into a lexicographic rank.

When to use this tool

Return a zero-based exact lexicographic permutation rank as decimal integer text; no floating factorial approximation.

Prepare the supplied permutation fields for this operation.
Compare the complete example with the representation used by your own input before carrying the result into another step.

How to use the tool and how it works

Enter permutation. Use the exact text syntax, units and index conventions explained below.
Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
Read result.rank. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.

Validate a permutation of zero-based indexes, compute each smaller-right count and sum digit[j] times factorial(length minus j minus one). Return decimal text.^[1]

Input field	Meaning and representation	Accepted input
permutation	permutation.	List of finite numbers; at most 128 entries

Output field	Interpretation
result	Object containing the named outputs below.
result.rank	The rank value under the stated formula or text convention.

json

{
  "input": {
    "permutation": [
      2,
      0,
      1
    ]
  },
  "output": {
    "result": {
      "rank": "4"
    }
  }
}

Limitations and assumptions

The exact rank starts at zero. Duplicate values, arbitrary labels and ranks from another convention are not supported.
Except in the diagnostic validator, a permutation must contain each integer from zero through length minus one exactly once. Lists contain at most 128 entries.
All declared input fields are required and unexpected top-level fields are rejected. Numbers must be finite; lists are limited to 128 entries and ordinary text fields to 8,192 UTF-16 code units, with tighter limits where stated.
JSON text inputs must parse within bounded nesting and size limits. Numeric outputs outside the safe finite range are rejected; large exact integer results use decimal strings. Indexes and half-open spans are zero-based unless a rank is explicitly described as one-based.

Alternative or Complementary methods

Keep the index-to-image convention and composition direction explicit. A rank, cycle table, reorder-index list and inverse map encode different operations even when their entries are all integers.

References

Tool behavior and example: Permutation Lexicographic Rank — tool.io

Similar or alternative tools

Permutation Lexicographic Unrank
Unrank a zero-based lexicographic index permutation from exact integer text and an explicit bounded size.
Next Lexicographic Permutation
Return the next lexicographic index permutation without wraparound; null marks the boundary of the permutation order.