Hankel determinants of sums of consecutive weighted Schr\"{o}der numbers

TL;DR

This paper derives recurrence formulas for Hankel determinants of sums involving weighted Schr"{o}der numbers, using combinatorial lattice path models, expanding understanding of their algebraic and combinatorial properties.

Contribution

It introduces new recurrence relations for Hankel determinants of sums of weighted Schr"{o}der numbers, connecting algebraic formulas with combinatorial lattice path models.

Findings

Recurrence formulas for Hankel determinants involving weighted Schr"{o}der numbers.

Combinatorial interpretations via lattice path models.

Extension of determinant analysis to weighted Schr"{o}der paths.

Abstract

For a real number $t$, let $r_\ell(t)$ be the total weight of all $t$-large Schr\"{o}der paths of length $\ell$, and $s_\ell(t)$ be the total weight of all $t$-small Schr\"{o}der paths of length $\ell$. For constants $\alpha, \beta$, in this article we derive recurrence formulae for the determinats of the Hankel matrices $\det_{1\le i,j\le n} (\alpha r_{i+j-2}(t) +\beta r_{i+j-1}(t))$, $\det_{1\le i,j\le n} (\alpha r_{i+j-1}(t) +\beta r_{i+j}(t))$, $\det_{1\le i,j\le n} (\alpha s_{i+j-2}(t) +\beta s_{i+j-1}(t))$, and $\det_{1\le i,j\le n} (\alpha s_{i+j-1}(t) +\beta s_{i+j}(t))$ combinatorially via suitable lattice path models.