What Permutation Rank Means
A permutation is an arrangement of elements in a specific order. When you list all permutations of the same elements and sort that list using a consistent rule, every permutation becomes an item in an ordered sequence. The rank of a permutation is its position (index) inside that ordered sequence.
The most common ranking rule is lexicographic order, which behaves like dictionary sorting. Compare two permutations element-by-element starting from the first position. The one with the smaller first element is earlier. If the first elements match, compare the second elements, and so on. To make that comparison meaningful, the elements themselves must have a defined order, typically the sorted order of the element set.
This Permutation Rank Calculator computes lexicographic rank for distinct elements, supports a multiset version for duplicates, and can also do the reverse operation called unranking, where you generate the permutation that corresponds to a chosen rank.
Why Permutation Ranking Is Useful
Ranking and unranking create a bridge between two worlds: the world of arrangements (permutations) and the world of integers (indices). That bridge is valuable in many contexts. In programming, it lets you map a permutation to a unique integer key for hashing, caching, or deterministic iteration. In combinatorics, it helps you count and compare arrangements without writing out the entire list. In algorithms, it supports generating permutations in a specific order or jumping directly to a particular permutation by index.
Common practical uses include:
- Iterating permutations in a reproducible sequence
- Encoding permutations as numbers for storage or indexing
- Random sampling by choosing a random rank and unranking it
- Testing and debugging permutation-based algorithms
- Studying ordering, inversions, and combinatorial structures
Lexicographic Order for Custom Elements
Lexicographic order depends on how you compare elements. For numeric elements, the “natural” order is 1<2<3<… . For symbols like A, B, C, D, the order is alphabetical. For custom tokens such as names or labels, the order can be the sorted order of those tokens or a custom order you choose.
In the Rank tab, you can compute rank using the automatically sorted element set derived from your permutation, or you can supply a custom element set. Custom sets are useful when you need a specific ordering (for example, a domain-specific order that is not alphabetical).
Distinct-Element Rank and the Lehmer Code
For permutations of distinct elements, an efficient ranking method uses the Lehmer code. At each position i, count how many unused elements smaller than the current element remain. That count becomes a digit. Those digits are not ordinary base-10 digits; they form a mixed-base representation known as the factorial number system, or factoradic.
The reason factorial bases appear is simple: if there are n elements, there are (n−1)! permutations for each fixed first element. Then, after fixing the first element, there are (n−2)! permutations for each fixed second element, and so on. Those factorial “block sizes” become the weights in the rank calculation.
rank = Σ (count of smaller unused at position i) · (n−1−i)!
This calculator shows those counts and factorial weights in a step table so you can see exactly how the rank is constructed.
Factoradic: The Number System Behind Unranking
The same structure that ranks permutations also un-ranks them. If you write a rank in factoradic form, you get a sequence of digits where the rightmost digit is in base 1, the next digit is in base 2, then base 3, and so on. Those digits tell you which element to pick from a remaining sorted list at each step.
Unranking works like this:
- Start with a sorted list of elements.
- Convert the rank (0-based) into factoradic digits.
- Use each digit as an index into the remaining elements, removing the chosen element each time.
The Unrank tab implements this process and shows the factoradic digits and the element chosen at every step. It also supports generating permutations of 1..n for convenience, which is a common math and programming convention.
0-Based vs 1-Based Rank
Rank conventions vary. In many programming languages and algorithm references, ranks are 0-based: the first permutation has rank 0. In many math contexts, it is natural to label the first permutation as 1. This tool supports both by letting you specify the rank base for input while showing both representations in the results.
The conversion is straightforward: rank(1-based) = rank(0-based) + 1. For unranking with 1-based input, the tool subtracts 1 before generating the permutation.
Ranking Permutations With Duplicate Elements
When elements repeat, not all arrangements are distinct. For example, A,A,B,C has fewer unique permutations than 4! because swapping the A’s does not create a new sequence. These cases are handled by multiset permutations, counted by a multinomial-style formula:
total = n! / (k1! · k2! · …)
Multiset ranking is more subtle than distinct-element ranking. When you place a smaller element at the current position, the number of distinct permutations that follow depends on the remaining counts of each repeated element. The Rank with Duplicates tab computes rank by summing the number of valid completions for each smaller choice at each position, using exact integer arithmetic.
This is especially useful for ranking anagrams, repeated-letter strings, multiset schedules, and any “arrangements with repeats” problem where you want a unique index among distinct permutations.
How to Use the Permutation Rank Calculator
Rank a distinct permutation
- Enter your permutation as comma-separated or space-separated values.
- Use sorted order by default, or provide a custom element set if you need a specific ordering.
- Choose 0-based or 1-based rank base.
- Click Calculate Rank to see rank, total permutations, Lehmer code, and the step table.
Generate a permutation from a rank
- Choose 1..n or a custom element set.
- Enter the rank and choose base.
- Click Generate Permutation to see the factoradic digits and pick sequence.
Rank a permutation with duplicates
- Enter a permutation that may repeat elements (like A,A,B,C).
- Select rank base and click Calculate Multiset Rank.
- Review the added-permutations breakdown and total distinct permutations.
Interpreting Results and Avoiding Common Errors
The most important detail is the element ordering. Lexicographic rank is defined relative to a specific ordering of the element set. If you change the ordering, the rank changes, even though the permutation itself is the same. When in doubt, use the default sorted ordering for clarity and reproducibility.
Another common issue is mixing distinct and duplicate logic. If your permutation contains repeated elements, distinct-element rank does not apply. Use the duplicates tab so the tool counts only distinct permutations.
Finally, keep scale in mind. The number of permutations grows factorially, so total counts become enormous quickly. This calculator uses exact integer arithmetic, but extremely large n can still become slow to display and reason about. The step tables and exports are designed to keep the computation transparent, even when numbers are large.
Where Permutation Rank Shows Up in Real Work
Permutation rank appears in algorithm design (generating a specific permutation without enumerating all previous ones), in testing frameworks (deterministic permutation generation), in combinatorial search (mapping states to indices), and in probability and statistics (enumerating ordered outcomes). In puzzle and game contexts, ranking provides a compact way to store and compare states. In cryptography and security discussions, ranking helps describe keyspaces when the key is a permutation rather than a string.
FAQ
Permutation Rank Calculator – Frequently Asked Questions
Answers about lex order, 0-based vs 1-based rank, Lehmer code, factoradics, multiset permutations, and exporting steps.
The rank of a permutation is its index in an ordered list of all permutations. Most commonly, rank refers to lexicographic order (dictionary order) based on a sorted set of elements.
Lexicographic order sorts permutations the same way words are sorted in a dictionary: compare the first element, then the second, and so on. The element ordering is defined by the sorted element set you choose.
In programming, rank is often 0-based (first permutation has rank 0). In math and many textbooks, rank is often 1-based. This calculator shows both and lets you choose your input base.
A common method uses the Lehmer code, which counts how many unused smaller elements appear at each position. Those counts become digits in a factorial number system (factoradic) that maps directly to rank.
Unranking means generating the permutation that corresponds to a specific rank. Using factoradic digits, you repeatedly select an element from a list of remaining elements by index.
Yes. The calculator includes a multiset mode for duplicates (like A,A,B,C). It computes rank among distinct permutations using multinomial counting.
With duplicates, many arrangements collapse into the same permutation. Multiset ranking counts only distinct permutations, so the ordering and total count differ from the n! distinct-element case.
The Lehmer code is a sequence where each digit tells how many unused elements smaller than the current element appear to the right. It forms the factoradic representation of the permutation’s rank.
Yes. Each mode can export its step table to CSV so you can audit the computation or use it in a spreadsheet.