Updated Combinatorics & Order Tool

Permutation Rank Calculator

Compute the lexicographic rank of a permutation, recover the permutation from a given rank, handle repeated elements, and find the next or previous permutation in order. Ideal for combinatorics, algorithm design and contest problems.

Lexicographic Rank Unranking Multisets Next / Previous Permutation

Permutation Rank & Unranking Suite

This Permutation Rank Calculator works with both distinct elements and multisets with repeated elements. It lets you compute lexicographic rank, unrank from an index, find the next and previous permutations and even rank under a custom user-defined order. All calculations use exact integer arithmetic within practical size limits.

This tab assumes all elements are distinct and ranks the permutation in standard lexicographic order. The smallest permutation under this order has rank \(0\) and the largest has rank \(n!-1\).

Example inputs: 1 3 2 4, a b c d, A,C,B,D. The calculator sorts the elements to define the first permutation in lexicographic order.

Given a set of distinct elements and a rank \(r\) with \(0 \le r < n!\), this tab recovers the permutation with that lexicographic rank.

The elements are sorted to determine lexicographic order. The permutation with rank 0 is the sorted sequence.

This tab ranks permutations of a multiset, where some elements may repeat. The rank is defined among all distinct permutations in lexicographic order.

Example: 1 1 2 2, a a b c. The tool considers all distinct permutations of the multiset and returns the 0-based and 1-based rank.

Given the multiset of elements and a rank \(r\), this tab returns the \(r\)-th distinct permutation in lexicographic order.

The calculator first identifies the multiset type counts, then uses a counting argument to reconstruct the permutation with rank \(r\), if it exists.

This tab finds the next and previous permutations of a given sequence in lexicographic order. The first permutation has no previous permutation and the last has no next permutation.

Sometimes the natural lexicographic order is defined by a custom alphabet or priority list. This tab ranks permutations under a user-specified element order.

The custom order defines which element is considered "smaller" when comparing permutations. All permutation elements must appear in the custom order list.

This tab moves forward in lexicographic order by \(k\) steps. A step means moving to the next permutation in lexicographic order. The calculator works with distinct elements.

If \(k\) is large enough to pass the last permutation, the tool stops at the final permutation.

This tab tells you how many permutations remain in lexicographic order starting from the current permutation (inclusive and exclusive). It assumes distinct elements.

This tab summarizes factorial growth and practical limits for ranking permutations in a browser environment.

For large \(n\), the number of permutations \(n!\) grows extremely quickly, which limits how far ranking and unranking can go using standard floating-point arithmetic.

Permutation Rank Calculator – Understanding Lexicographic Order and Unranking

The Permutation Rank Calculator on MyTimeCalculator is a focused tool for working with permutations in lexicographic order. It helps you answer questions such as: “What is the position of this permutation in sorted order?”, “What permutation occupies rank \(r\)?”, “What comes next or just before?” and “How do repeated elements change the counting?”.

Ranking and unranking algorithms are standard in combinatorics, algorithm design, cryptography, coding theory and competitive programming. By turning permutations into integers and back, they provide a bridge between combinatorial structures and numeric indices.

1. Lexicographic Order on Permutations

Lexicographic order on permutations is defined by sorting the underlying elements and then comparing sequences from left to right. Given a list of distinct elements, the first permutation in lexicographic order is the sorted sequence, and the last is the reverse of the sorted sequence. The number of permutations of \(n\) distinct elements is

\[ n! = 1 \cdot 2 \cdot 3 \cdots n. \]

If a permutation \(\pi\) has rank \(r\) in this order, then exactly \(r\) permutations come before it and \(n! - 1 - r\) permutations come after it. The calculator reports both the 0-based rank (starting at 0) and the 1-based rank (starting at 1), since both conventions are common in textbooks and contest problems.

2. How the Rank of a Permutation Is Computed

For distinct elements, ranking is based on counting how many permutations begin with a prefix that is smaller in lexicographic order. Suppose the sorted elements are

\[ a_1 < a_2 < \dots < a_n, \]

and we want the rank of a permutation \(\pi = (\pi_1,\pi_2,\dots,\pi_n)\). For each position, we:

  • Count how many unused elements are smaller than \(\pi_i\).
  • Multiply this count by \((n-i)!\), the number of permutations of the remaining positions.
  • Add this contribution to the rank and then mark \(\pi_i\) as used.

The total gives the 0-based lexicographic rank. Unranking reverses this logic: it repeatedly divides the rank by descending factorials to pick the appropriate element at each position.

3. Multisets and Repeated Elements

When elements may repeat, the total number of distinct permutations is smaller. If a multiset has \(n\) total elements with type counts \(a_1,a_2,\dots,a_k\) such that \(a_1 + \dots + a_k = n\), the count of distinct permutations is

\[ \frac{n!}{a_1! a_2! \cdots a_k!}. \]

Ranking under multiset permutations still follows the idea of counting how many permutations start with lexicographically smaller prefixes, but each time we hypothetically place a smaller symbol, the number of completions is computed using the multiset permutation formula with one fewer copy of that symbol.

4. Next and Previous Permutations

The classic algorithm for the next permutation in lexicographic order works in three steps:

  1. Scan from the right to find the longest non-increasing suffix.
  2. Swap the pivot element just before the suffix with the smallest element in the suffix that is larger than it.
  3. Reverse the suffix to get the smallest possible tail.

Applying this procedure repeatedly generates all permutations in lexicographic order. The calculator uses this method to find both the next and previous permutations, and also to move forward by a given number \(k\) steps by combining ranking and unranking.

5. Custom Orderings and Applications

In some settings, elements do not follow the usual numeric or alphabetical order. For example, card suits may be ordered as \(\text{♣}, \text{♦}, \text{♥}, \text{♠}\) or priorities may be given in a custom list. In that case, comparisons between permutations should respect the custom order.

The Permutation Rank Calculator lets you specify a custom order list and then ranks permutations by translating each element into a code based on its position in that list. This is useful in card games, sorting tasks, custom encodings and algorithms that rely on non-standard ordering rules.

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Permutation Rank Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about lexicographic rank, unranking, multisets and next or previous permutations in this calculator.

In 0-based ranking, the first permutation in order has rank 0, the next has rank 1, and so on up to \(n!-1\). In 1-based ranking, the first permutation has rank 1 and the last has rank \(n!\). Many algorithms naturally use 0-based rank, while some textbooks and contest statements prefer 1-based rank. The calculator reports both so you can match the convention in your problem.

The value of \(n!\) grows very quickly. For modest n, the factorial already exceeds billions or trillions, which can still be handled numerically but not listed explicitly. Beyond certain thresholds, floating-point arithmetic can no longer represent all intermediate values reliably. The calculator enforces practical limits to keep results consistent and avoid numerical overflow or underflow in a browser environment.

Yes. The multiset tabs handle repeated elements explicitly. They treat permutations that differ only by swapping identical items as the same arrangement and compute ranks among distinct permutations of the multiset. For purely distinct elements, the dedicated distinct tabs are simpler and slightly faster.

For the standard tabs, elements are treated as strings and sorted using their natural order, which is similar to dictionary order. For example, the order of the symbols might be A < B < C or 1 < 2 < 10 depending on how they are written. If you need a specific ordering that does not match the default, you can use the custom order tab to explicitly define the precedence of each symbol.

Many interview and contest problems involve generating the next permutation, mapping between permutations and ranks, or working with permutations that include repeated elements. This calculator mirrors the underlying ranking and unranking procedures used in code, so you can use it to test examples, verify your implementations and better understand the structure of permutation spaces.

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