Hankel determinants of sums of consecutive weighted Schröder numbers

Sen Peng Eu, Tsai Lien Wong, Pei Lan Yen

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)2285-2299
Number of pages15
JournalLinear Algebra and Its Applications
Volume437
Issue number9
DOIs
Publication statusPublished - 2012 Nov 1

Fingerprint

Hankel Determinant
Consecutive
Path
Lattice Paths
Hankel Matrix
Generating Function
Explicit Formula
Determinant

Keywords

  • Combinatorial methods
  • Hankel determinants
  • Non-intersecting lattice paths
  • Schröder numbers

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

Cite this

Hankel determinants of sums of consecutive weighted Schröder numbers. / Eu, Sen Peng; Wong, Tsai Lien; Yen, Pei Lan.

In: Linear Algebra and Its Applications, Vol. 437, No. 9, 01.11.2012, p. 2285-2299.

Research output: Contribution to journalArticle

Eu, Sen Peng ; Wong, Tsai Lien ; Yen, Pei Lan. / Hankel determinants of sums of consecutive weighted Schröder numbers. In: Linear Algebra and Its Applications. 2012 ; Vol. 437, No. 9. pp. 2285-2299.
@article{2620313d60164ff4a0b66bc8e375ce2d,
title = "Hankel determinants of sums of consecutive weighted Schr{\"o}der numbers",
abstract = "We consider weighted large and small Schr{\"o}der 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 Schr{\"o}der 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.",
keywords = "Combinatorial methods, Hankel determinants, Non-intersecting lattice paths, Schr{\"o}der numbers",
author = "Eu, {Sen Peng} and Wong, {Tsai Lien} and Yen, {Pei Lan}",
year = "2012",
month = "11",
day = "1",
doi = "10.1016/j.laa.2012.05.024",
language = "English",
volume = "437",
pages = "2285--2299",
journal = "Linear Algebra and Its Applications",
issn = "0024-3795",
publisher = "Elsevier Inc.",
number = "9",

}

TY - JOUR

T1 - Hankel determinants of sums of consecutive weighted Schröder numbers

AU - Eu, Sen Peng

AU - Wong, Tsai Lien

AU - Yen, Pei Lan

PY - 2012/11/1

Y1 - 2012/11/1

N2 - 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.

AB - 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.

KW - Combinatorial methods

KW - Hankel determinants

KW - Non-intersecting lattice paths

KW - Schröder numbers

UR - http://www.scopus.com/inward/record.url?scp=84864756223&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864756223&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2012.05.024

DO - 10.1016/j.laa.2012.05.024

M3 - Article

AN - SCOPUS:84864756223

VL - 437

SP - 2285

EP - 2299

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 9

ER -