[Paper Review] Permutation sign under the Robinson-Schensted-Knuth correspondence

Astrid Reifegerste|ArXiv.org|Sep 16, 2003

Advanced Combinatorial Mathematics5 references4 citations

TL;DR

This paper establishes a formula for computing the sign of a permutation directly from the Robinson-Schensted-Knuth (RSK) correspondence by analyzing the insertion and recording tableaux. It shows that the sign equals the product of the signs of the two tableaux and the parity of the total length of even-indexed rows in the insertion tableau, providing a simple proof of Stanley's conjecture on imbalances of hook-shaped tableaux and refining sign distributions in pattern-avoiding permutations.

ABSTRACT

We show how the sign of a permutation can be deduced from the tableaux induced by the permutation under the Robinson-Schensted-Knuth correspondence. The result yields a simple proof of a conjecture on the squares of imbalances raised by Stanley.

Motivation & Objective

• To determine the sign of a permutation using only the tableaux produced by the RSK correspondence.
• To provide a generalization of Beissinger’s algorithm for involutions to arbitrary permutations.
• To prove Stanley’s conjecture regarding the squares of imbalances in standard Young tableaux of hook shape.
• To analyze the joint distribution of sign and longest increasing subsequence in pattern-avoiding permutations.

Proposed method

• Leverages the symmetry of the RSK correspondence: the recording tableau of a permutation is the insertion tableau of its inverse.
• Applies Knuth’s equivalence relation to extend results from involutions to general permutations via elementary transformations.
• Defines the sign of a tableau as (−1) raised to the number of inversions in its row word.
• Introduces the concept of even-indexed row length parity as a key component in computing the permutation sign.
• Uses reverse RSK insertion to reconstruct permutations from tableaux and identify descent structures.
• Applies known results on hook-shaped tableaux and imbalances to derive generating functions for sign-weighted distributions.

Experimental results

Research questions

• RQ1 How can the sign of a permutation be computed directly from the RSK correspondence tableaux?
• RQ2 What is the relationship between the sign of a permutation and the structure of its insertion and recording tableaux?
• RQ3 Can the sign be determined using only the shape and content of the tableaux, independent of the permutation’s cycle structure?
• RQ4 How does the sign distribution behave in permutations avoiding the patterns 213 and 231?
• RQ5 Does the formula for the sign of a permutation via tableaux provide a proof of Stanley’s conjecture on imbalances of hook-shaped tableaux?

Key findings

• The sign of a permutation is equal to the product of the signs of its insertion and recording tableaux and the parity of the total length of even-indexed rows in the insertion tableau.
• The result provides a new, simple proof of Stanley’s conjecture on the squares of imbalances in standard Young tableaux of hook shape.
• For permutations avoiding both 213 and 231, the generating function for sign-weighted longest increasing subsequence is given by $ q(q+1)^{ floor(n-1)/2 floor}(q-1)^{ floor n/2 floor} $.
• When $ n $ is odd, the number of even and odd permutations in the set of 213/231-avoiding permutations is equal, as the generating function simplifies to $ q(q^2 - 1)^{(n-1)/2} $.
• When $ n $ is even, the generating function becomes $ q(q-1)(q^2 - 1)^{n/2 - 1} $, reflecting a sign imbalance in the distribution.
• The paper establishes a bijection between {213,231}-avoiding permutations and standard Young tableaux of hook shape, confirming $ |A_n| = 2^{n-1} $ via RSK symmetry.