We study generalized small Schröder paths in the sense of arbitrary sizes of steps. A generalized small Schröder path is a generalized lattice path from $(0,0)$ to $(2n,0)$ with the step set of $\{(k,k), (l,-l), (2r,0)\, |\, k,l,r \in {\bf P}\}$, where ${\bf P}$ is the set of positive integers, which never goes below the $x$-axis, and with no horizontal steps at level 0. We find a bijection between 5-colored Dyck paths and generalized small Schröder paths, proving that the number of generalized small Schröder paths is equal to $\sum_{k=1}^{n} N(n,k)5^{n-k}$ for $n\geq 1$.
@article{10_37236_4827,
author = {JiSun Huh and SeungKyung Park},
title = {Generalized small {Schr\"oder} numbers},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {3},
doi = {10.37236/4827},
zbl = {1327.05017},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4827/}
}
TY - JOUR
AU - JiSun Huh
AU - SeungKyung Park
TI - Generalized small Schröder numbers
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/4827/
DO - 10.37236/4827
ID - 10_37236_4827
ER -
%0 Journal Article
%A JiSun Huh
%A SeungKyung Park
%T Generalized small Schröder numbers
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/4827/
%R 10.37236/4827
%F 10_37236_4827
JiSun Huh; SeungKyung Park. Generalized small Schröder numbers. The electronic journal of combinatorics, Tome 22 (2015) no. 3. doi: 10.37236/4827
