[Paper Review] Generating permutations with a given major index

Vincent Vajnovszki|arXiv (Cornell University)|Feb 26, 2013

Coding theory and cryptography3 references3 citations

TL;DR

This paper presents the first constant amortized time (CAT) algorithm for generating permutations with a fixed major index, using a novel Gray code for subexcedant sequences that ensures consecutive permutations differ in at most three adjacent transpositions. The method leverages a co-lex order generation of bounded compositions, transforms it into a Gray code for subexcedant sequences, and applies a bijection to permutations, preserving the Gray code property through controlled transposition sequences.

ABSTRACT

In [S. Effler, F. Ruskey, A CAT algorithm for listing permutations with a given number of inversions, {\it I.P.L.}, 86/2 (2003)] the authors give an algorithm, which appears to be CAT, for generating permutations with a given major index. In the present paper we give a new algorithm for generating a Gray code for subexcedant sequences. We show that this algorithm is CAT and derive it into a CAT generating algorithm for permutations with a given major index.

Motivation & Objective

• To develop a constant amortized time (CAT) algorithm for generating permutations with a fixed major index.
• To construct a Gray code for subexcedant sequences of fixed weight, where consecutive sequences differ in at most three adjacent positions by bounded amounts.
• To ensure the Gray code property is preserved under the bijection between subexcedant sequences and permutations, enabling efficient generation of permutations with fixed major index.
• To demonstrate that the Gray code structure is essential and cannot be replaced by a simpler lexicographic order without losing efficiency.

Proposed method

• Generate bounded compositions in co-lex order using a recursive algorithm with increasable positions to avoid redundant nodes in the generation tree.
• Transform the co-lex order into a Gray code for subexcedant sequences by reordering the generation process and specializing to the bounding sequence $\boldsymbol{b} = 01\ldots(n-1)$.
• Optimize the Gray code generation into an efficient version, Gen2_Gray, which tracks the rightmost differing position and cumulative sum for fast updates.
• Define a transposition procedure, transp, that updates the permutation based on differences in subexcedant sequences, using a position-based transformation with $\phi_j$ functions.
• Use a bijection $\psi$ from subexcedant sequences to permutations (McMahon codes) to map the Gray code on sequences to a Gray code on permutations.
• Apply the Update_Perm procedure to compute the minimal sequence of transpositions transforming one permutation into the next, based on local differences in the subexcedant sequence.

Experimental results

Research questions

• RQ1 Can a constant amortized time algorithm be constructed for generating permutations with a fixed major index?
• RQ2 Is it possible to design a Gray code for subexcedant sequences of fixed weight such that consecutive sequences differ in at most three adjacent positions by bounded amounts?
• RQ3 Does the Gray code structure on subexcedant sequences preserve the property that permutations differ by few transpositions under the McMahon code bijection?
• RQ4 Why is the Gray code structure essential in this construction, and why can't a lexicographic order achieve the same efficiency?

Key findings

• The proposed algorithm generates all permutations of length $n$ with major index $k$ in constant amortized time per permutation.
• The Gray code for subexcedant sequences ensures that consecutive sequences differ in at most three adjacent positions, with changes bounded by $\pm1$ or $\pm2$.
• Consecutive permutations generated via the bijection differ by at most three transpositions, a key structural property enabling efficient generation.
• The algorithm is derived by transforming a co-lex order generation of bounded compositions into a Gray code, then specializing to subexcedant sequences and optimizing the generation process.
• The transposition-based update mechanism, implemented via the Update_Perm procedure, correctly maps local differences in subexcedant sequences to global permutation changes.
• Numerical evidence shows that relaxing the Gray code condition (e.g., allowing non-adjacent differences) breaks the transposition efficiency in the permutation space.