bibkey: cigler2022motzkin authors: Johann Cigler year: 2022 title: “Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths” doi: 10.48550/arXiv.2204.09910 url: https://arxiv.org/abs/2204.09910v4 claim: “Conjectures on the generating functions of Hankel determinants of Motzkin path polynomials: reduced denominators and numerator degrees for the columns of the Motzkin triangle (Conjectures 1.2, 1.3) and for a boundary weight s at height zero (Conjecture 2.1).” strata_touched:

• D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankel
• D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumn license: citation-only triage: anchor

Cigler, Hankel determinants of polynomials related to Motzkin paths

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; in Section 2 the horizontal steps at height 0 carry a separate weight s. The paper studies the Hankel determinants d_m^{(k)}(n,t) = det(M_{m+i+j,k}(t))_{0≤i,j<n} and their generating functions D_m^{(k)}(x,t), which are rational in x.

Conjecture 1.2 gives a common denominator ∏j A{k,(k+1)(m−2j)}(x,t)^{binom(m,j)}; Conjecture 1.3 reduces the exponents to 1 + j(m − j) and gives the exact x-degree of the numerator. Conjecture 2.1 states the analogous reduced denominator and numerator degree binom(m+1,3) + 1 for column 0 with boundary weight s. The case k = 0 of the weaker denominator follows from Krattenthaler’s Corollary 9, recorded in the paper as Theorem 1.1.

The module D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankel proves Conjecture 2.1.

Verified locator

DOI: 10.48550/arXiv.2204.09910

URL: https://arxiv.org/abs/2204.09910v4

• Locator: Section 1, equations (1.28)–(1.30), Conjectures 1.2 and 1.3.
• Locator: Section 2, equation (2.3), Conjecture 2.1.