[Paper Review] A recurrence formula for Jack connection coefficients

A. L. Kanunnikov, Ekaterina A. Vassilieva|arXiv (Cornell University)|Sep 15, 2014

Random Matrices and Applications11 references3 citations

TL;DR

This paper establishes a recurrence formula for Jack connection coefficients when two partitions are equal to (n), enabling efficient computation and proving the Matchings-Jack conjecture in this case. Using Lasalle's framework for Jack symmetric functions, the authors derive a polynomial structure in β = α−1 with non-negative integer coefficients, validated via bijective proofs for α ∈ {1,2} and extended to more general coefficient families.

ABSTRACT

This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack symmetric functions. First introduced by Goulden and Jackson (1996) these numbers indexed by three partitions of a given integer $n$ and the Jack parameter $α$ are defined as the coefficients in the power sum expansion of the Cauchy sum for Jack symmetric functions. While very little is known about them, examples of computations for small values of $n$ tend to show that the nice properties of the special cases $α=1$ (connection coefficients of the class algebra) and $α= 2$ (connection coefficients of the double coset algebra) extend to general $α$. Goulden and Jackson conjectured that Jack connection coefficients are polynomials in $β= α-1$ with non negative integer coefficients given by some statistics on matchings on a set of $2n$ elements, the so called Matchings-Jack conjecture. In this paper we look at the case when two of the integer partitions are equal to the single part $(n)$ and use a framework by Lasalle (2008) for Jack symmetric functions to show that the coefficients satisfy a simple recurrence formula that makes their computation very effective and allow a better understanding of their properties. In particular we prove the Matchings-Jack conjecture in this case. Furthermore, we provide a bijective proof of the recurrence formula for $α\in \{1,2\}$ using the combinatorial interpretation of the coefficients for these specific values of the Jack parameter. Finally we exhibit the polynomial properties of more general coefficients where the two single part partitions are replaced by an arbitrary number of integer partitions either equal to $(n)$ or $[1^{n-2}2]$.

Motivation & Objective

• To investigate Jack connection coefficients in the special case where two of the three partitions are equal to (n), a configuration with rich combinatorial structure.
• To establish a recurrence formula that simplifies the computation of these coefficients and reveals their underlying polynomial structure in β = α−1.
• To prove the Matchings-Jack conjecture in this specific case, confirming that coefficients are polynomials in β with non-negative integer coefficients.
• To provide a bijective proof of the recurrence for α = 1 and α = 2, linking the algebraic formula to combinatorial matchings.
• To generalize the polynomial properties to coefficients involving partitions (n) and [1^{n−2}2], extending the scope of the conjecture.

Proposed method

• Leverages Lasalle’s framework for Jack symmetric functions to derive a recurrence for Jack connection coefficients when two partitions are (n).
• Introduces a weight function wt_λ(δ) on matchings δ to encode the recurrence, depending on the starting vertex in non-bipartite matchings.
• Uses the operator Δ_l(α) = [D(α), [..., [D(α), p_1/α]...]] to generate generating series and relate coefficients across partition sizes.
• Applies the operator D(α) = α^{-1} ∑_k (k-1)α^{k-1} p_k ⊥ to manipulate power sum expansions and extract coefficient polynomials.
• Employs the duality relation α^{-ℓ(λ)} z_λ^{-1} a_λ^{l,r}(α) = (1/αn!) ∑ g_i α^i to prove symmetry and polynomiality of coefficients.
• Validates the recurrence via bijective interpretation for α = 1 and α = 2 using the known combinatorial models of class algebra and double coset algebras.

Experimental results

Research questions

• RQ1 Do Jack connection coefficients with two partitions equal to (n) satisfy a recurrence formula that simplifies their computation and reveals polynomial structure in β = α−1?
• RQ2 Is the Matchings-Jack conjecture valid for Jack coefficients when two partitions are (n), i.e., are they polynomials in β with non-negative integer coefficients?
• RQ3 Can the recurrence formula be given a bijective interpretation for α = 1 and α = 2 using known combinatorial models of matchings and symmetric functions?
• RQ4 What is the degree and symmetry of the polynomial structure of more general Jack connection coefficients involving partitions (n) and [1^{n−2}2]?
• RQ5 How do the coefficients a_λ^{l,r}(α) behave under duality and what constraints does this impose on their polynomial form?

Key findings

• The recurrence formula for Jack connection coefficients with two (n) partitions allows for efficient computation and confirms the Matchings-Jack conjecture in this case.
• The coefficients a_{n,n}^{n}(β+1) are polynomials in β with non-negative integer coefficients, and for n=3, a_{3,3}^{3}(β+1) = 2β² + β + 1.
• For α = 1 and α = 2, the recurrence is proven bijectively using matchings on 2n elements, linking algebraic coefficients to combinatorial statistics.
• The generating series Γ_n^l satisfies Γ_n^l = (1/n) Δ_l(α)(Γ_{n-1}^l), establishing a recursive structure across partition sizes.
• The coefficients a_λ^{l,r}(α) are polynomials in α of degree at most (n−1)(l−1) + r, with symmetric coefficients under the transformation α → α^{-1} up to sign.
• The duality relation [α^i] |C_λ| a_λ^{l,r}(α) = (−1)^{(l−1)(n−1)+r+ℓ(λ)−1} [α^{(l−1)(n−1)+r+ℓ(λ)−1−i}] |C_λ| a_λ^{l,r}(α) confirms the palindromic symmetry of the coefficient polynomials.