### 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 language | English |
---|---|

Pages (from-to) | 2285-2299 |

Number of pages | 15 |

Journal | Linear Algebra and Its Applications |

Volume | 437 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2012 Nov 1 |

### Fingerprint

### 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

*Linear Algebra and Its Applications*,

*437*(9), 2285-2299. https://doi.org/10.1016/j.laa.2012.05.024

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

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 437, no. 9, pp. 2285-2299. https://doi.org/10.1016/j.laa.2012.05.024

}

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 -