[Paper Review] A bijective proof of an unusual symmetric group generating function

Mike Zabrocki|ArXiv.org|Oct 19, 2003

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper presents a bijective proof of a symmetric group generating function identity involving a novel statistic, baj(σ), defined as the sum over descents of i(n−i). The authors construct a weight-preserving bijection between permutations σ ∈ Sₙ with σₙ = k and integer sequences (r₁,…,rₙ₋₁) with 0 ≤ rᵢ < i, such that baj(σ) − inv(σ) = ∑(n−i)rᵢ. This establishes the identity ∑_{σ:σₙ=k} q^{baj(σ)−inv(σ)} = ∏_{i=1}^{n−1} (1−q^{i(n−i)}) / (1−q^i), resolving a special case of a Weyl group generating function conjectured by Stembridge.

ABSTRACT

For $σ\in S_n$, let $D(σ) = \{i : σ_{i} &gt; σ_{i+1}\}$ denote the descent set of $σ$. The length of the permutation is the number of inversions, denoted by $inv(σ) = \big | \{(i,j) : i σ_j\} \big |$. Define an unusual quadratic statisitic by $baj(σ) = \sum_{i \in D(σ)} i (n-i)$. We present here a bijective proof of the identity $\sum_{{σ\in S_n} \atop {σ(n) = k}} q^{baj(σ) - inv(σ)} = \prod_{i=1}^{n-1} {1-q^{i (n-i)} \over {1-q^i}}$ where $k$ is a fixed integer.

Motivation & Objective

• To provide a bijective proof of a symmetric group generating function identity involving a quadratic statistic baj(σ) on permutations.
• To resolve a special case of a Weyl group generating function identity presented by John Stembridge for the A_{n−1} root system.
• To establish a weight-preserving bijection between permutations with fixed last element and integer sequences with bounded entries.
• To demonstrate that the difference baj(σ) − inv(σ) equals a linear form in the sequence (r₁,…,rₙ₋₁), enabling the generating function identity.

Proposed method

• Define the statistic baj(σ) = ∑_{i∈D(σ)} i(n−i), where D(σ) is the descent set of σ.
• Represent permutations via the sequence (v₁,…,vₙ), where vᵢ counts elements ≤ σᵢ among the first i positions.
• Construct a bijection by defining rᵢ = iχ(vᵢ₊₁ ≤ vᵢ) + vᵢ₊₁ − vᵢ − 1 for each i = 1,…,n−1.
• Show that the map from σ to (r₁,…,rₙ₋₁) is invertible when σₙ = k is fixed, using recursive reconstruction of the v-sequence.
• Prove that baj(σ) − inv(σ) = ∑_{i=1}^{n−1} (n−i)rᵢ by algebraic manipulation using the identity inv(σ) = (n+1 choose 2) − ∑vᵢ.
• Verify that the generating function over permutations with σₙ = k matches the product ∏_{i=1}^{n−1} (1−q^{i(n−i)}) / (1−q^i) via the bijection and weight preservation.

Experimental results

Research questions

• RQ1 Can a bijective proof be constructed for the generating function identity ∑_{σ:σₙ=k} q^{baj(σ)−inv(σ)} = ∏_{i=1}^{n−1} (1−q^{i(n−i)}) / (1−q^i)?
• RQ2 What is the combinatorial interpretation of the statistic baj(σ) − inv(σ) in terms of permutation structure?
• RQ3 Is there a weight-preserving bijection between permutations with fixed last element and integer sequences (r₁,…,rₙ₋₁) with 0 ≤ rᵢ < i?
• RQ4 How can the v-sequence (v₁,…,vₙ) be reconstructed from the r-sequence and the value vₙ = k?
• RQ5 Does the identity hold for all n and k ∈ {1,…,n}, and can it be derived from the A_{n−1} root system generating function?

Key findings

• The generating function ∑_{σ:σₙ=k} q^{baj(σ)−inv(σ)} equals ∏_{i=1}^{n−1} (1−q^{i(n−i)}) / (1−q^i), independent of k.
• The bijection between permutations with σₙ = k and sequences (r₁,…,rₙ₋₁) with 0 ≤ rᵢ < i is weight-preserving with respect to the statistic baj(σ) − inv(σ).
• The statistic baj(σ) − inv(σ) is equal to ∑_{i=1}^{n−1} (n−i)rᵢ under the defined bijection.
• The v-sequence (v₁,…,vₙ) can be reconstructed from (r₁,…,rₙ₋₁) and vₙ = k via recursive formulas based on the indicator χ(vᵢ₊₁ ≤ vᵢ).
• The number of permutations with σₙ = k is equal to the number of such sequences (r₁,…,rₙ₋₁), confirming the bijection is size-preserving.
• The identity generalizes to the full symmetric group via ∑_{σ∈Sₙ} q^{baj(σ)−inv(σ)} = n × ∏_{i=1}^{n−1} (1−q^{i(n−i)}) / (1−q^i), as a consequence of the fixed-k case.