Hankel determinants of sums of consecutive weighted Schröder numbers

• Published in: Linear algebra and its applications 2012-11, Vol.437 (9), p.2285-2299
• Main Authors: Eu, Sen-Peng, Wong, Tsai-Lien, Yen, Pei-Lan
• Format: Article
• Language: English
• Subjects: Algebra; Combinatorial methods; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Hankel determinants; Linear and multilinear algebra, matrix theory; Mathematics; Non-intersecting lattice paths; Schröder numbers; Sciences and techniques of general use
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2012.05.024

We consider weighted large and small Schröder paths with up steps (1,1), down steps (1,-1) assigned the weight of 1 and with level steps (2,0) assigned the weight of t, where t is a real number. The weight of a path is the product of the weights of all its steps. Let rℓ(t) and sℓ(t) be the total weight of all weighted large and small Schröder paths from (0,0) to (2ℓ,0), respectively. For constants α, β, we derive the generating functions and the explicit formulae for the determinants of the Hankel matrices (αri+j-2(t)+βri+j-1(t))i,j=1n, (αri+j-1(t)+βri+j(t))i,j=1n, (αsi+j-2(t)+βsi+j-1(t))i,j=1n and (αsi+j-1(t)+βsi+j(t))i,j=1n combinatorially via suitable lattice path models.