TY - JOUR
T1 - Hankel determinants of sums of consecutive weighted Schröder numbers
AU - Eu, Sen Peng
AU - Wong, Tsai Lien
AU - Yen, Pei Lan
N1 - Funding Information:
Partially supported by National Science Council, Taiwan under Grant NSC 98-2115-M-390-002-MY3 (S.-P. Eu), NSC 99-2115-M-11∗Correspondingauthor.0-001-MY3 (T.-L.Wong). E-mail addresses: [email protected] (S.-P. Eu), [email protected] (T.-L. Wong), [email protected] (P.-L. Yen).
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
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U2 - 10.1016/j.laa.2012.05.024
DO - 10.1016/j.laa.2012.05.024
M3 - Article
AN - SCOPUS:84864756223
SN - 0024-3795
VL - 437
SP - 2285
EP - 2299
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 9
ER -