slug: cigler-motzkin-column-hankel bibkey: cigler2022motzkin doi: 10.48550/arXiv.2204.09910 url: https://arxiv.org/abs/2204.09910v4 triage: theorem motivation_gids:

• D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumn.result

Reduced Denominators for Hankel Determinants of Motzkin-Triangle Columns

Problem

Johann Cigler, Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths, arXiv:2204.09910v4, Section 1, Conjecture 1.3, equation (1.30). M_{n,k}(t) is the weighted number of Motzkin paths from (0,0) to (n,k) that never go below the axis, with up and down steps of weight 1 and horizontal steps of weight t; d_m^{(k)}(n,t) = det(M_{m+i+j,k}(t)){0≤i,j<n} and D_m^{(k)}(x,t) = Σ_n d_m^{(k)}(n,t) xⁿ. With e = (−1)^{binom(k+1,2)}, A{k,0} = 1 − e x^{k+1} and A_{k,r} = 1 − e L_r(t) x^{k+1} + x^{2(k+1)} for r > 0 (equation (1.28)), the conjecture states

D_m^{(k)}(x,t) = r_m^{(k)}(x,t) / ∏{j=0}^{⌊m/2⌋} A{k,(k+1)(m−2j)}(x,t)^{1+j(m−j)},

with deg_x r_m^{(k)} = binom(m+1,3) + k (binom(m,1) + binom(m,2) + binom(m,3)).

Motivation

The theorem D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumn.result establishes the identity for every column k ≥ 0 and every m ≥ 1 over ℤ[t], with the exact x-degree of the numerator.

Gap

Pre-registration issue 12625 records the literature screen: Krattenthaler’s Corollary 9 gives the weaker denominator for k = 0 (the paper’s Theorem 1.1); Cigler–Krattenthaler and Chern–Shi, arXiv:2608.27208, evaluate shifts m ≤ 2 and two residue classes for m ≤ k + 1, not the reduced exponents and numerator degree in general. This is a bounded negative finding.

Route

1. Specializing the boundary weight s to t in the proof of Conjecture 2.1 of the same paper (CiglerMotzkinHankel), a Christoffel-type formula writes d_m^{(k)}(n,t) as a determinant with k fixed Chebyshev nodes and m confluent zero nodes, with sign (−1)^{binom(k+1,2)}.
2. The two Binet branches and same-branch Vandermonde cancellation bound the polynomial parts by j(m − j), so the sequence satisfies the linear recurrence whose characteristic polynomial is the reversed denominator, with the x^{k+1} structure.
3. Negative-index determinants fix the exact x-degree of the numerator.

Falsifier

The statement would fail if the fixed nodes changed the sign e or the exponents, or if the first nonzero negative-index determinant vanished for some k.

Evidence

An independent referee implementation checked the identity symbolically in t for k ≤ 3, m ≤ 4 and for several integer t for k ≤ 4, m ≤ 5.

Triage

theorem; the statement is Conjecture 1.3 of arXiv:2204.09910v4, quantified over every k ≥ 0 and m ≥ 1.

• Proved (formalized): the reduced denominator and the exact numerator degree for every column k and every m ≥ 1.
• Proved (paper): Conjecture 1.2 of the same paper follows, since binom(m,j) ≥ 1 + j(m − j) makes the denominator of Conjecture 1.3 divide that of Conjecture 1.2.
• Open: whether the exponents 1 + j(m − j) are the reduced ones, i.e. whether the numerator is coprime to the denominator.

ASSUMED-UNVERIFIED

The literature screen is limited to the arXiv records named above, web and GitHub searches and the repository checks.