Toric g-contribution polynomials

Xiao studies the toric $g$-contribution polynomials $g_{n,j}(x)$, whose coefficients are built from binomial coefficients and Catalan numbers. Xiao's Conjecture 4.2 states that, for each $n = 2m + \varepsilon$ with $\varepsilon \in \{0, 1\}$, the row $(g_{n,0}, g_{n,1}, \dotsc, g_{n,\lfloor n/2\rfloor})$ is an interlacing sequence.

Theorem (Xiao's Conjecture 4.2). For all $m$ and $\varepsilon \leq 1$, the toric contribution row is an interlacing sequence.

The proof finds a common left interleaver for the normalized family: the terminating hypergeometric polynomials $R_d$ share a Jacobi-type interlacer. It follows that every strictly positive weighted sum of the normalized reversed contributions is real-rooted.

References

Q. Xiao, “The real-rootedness of the toric g-contribution polynomials,” arXiv:2609.01086 (2026). Common interleavers are described on the common interleaver page.

Definitions

Toric g-contribution polynomial

RealRooted.ParkingFunctions.ToricContribution.toricContributionsource

def toricContribution (m ε d : ℕ) : ℝ[X] :=
  (shiftedToricContribution m ε d).comp (X - 1)

Hypergeometric polynomials R_d

RealRooted.ParkingFunctions.ToricContribution.rPolynomialsource

def rPolynomial (m ε d : ℕ) : ℝ[X] :=
  ∑ k ∈ Finset.range (m + 1), monomial k (rCoeff m ε d k)

Theorems

Xiao's Conjecture 4.2

RealRooted.ParkingFunctions.ToricContribution.toricContributionRow_isInterlacingSeqsource

theorem toricContributionRow_isInterlacingSeq
    (m ε : ℕ) (hε : ε ≤ 1) :
    IsInterlacingSeq (toricContributionRow m ε)

The R_d have a common interleaver

RealRooted.ParkingFunctions.ToricContribution.normalizedRPolynomialFamily_hasCommonLeftInterleaversource

theorem normalizedRPolynomialFamily_hasCommonLeftInterleaver
    (m ε : ℕ) (hm : 0 < m) :
    HasCommonLeftInterleaver (normalizedRPolynomialFamily m ε)

Positive weighted sums are real-rooted

RealRooted.ParkingFunctions.ToricContribution.weightedNormalizedReversedContributionFamily_sum_splitssource

theorem weightedNormalizedReversedContributionFamily_sum_splits
    (m ε : ℕ) (hm : 0 < m) (hε : ε ≤ 1) (w : ℕ → ℝ)
    (hw : ∀ d, d ≤ m → 0 < w d) :
    (weightedNormalizedReversedContributionFamily m ε w).sum.Splits

Toric g-contribution polynomials

References

≔ Definitions

⊢ Theorems

Definitions

Theorems