Some bijections for lattice paths

TL;DR

This paper introduces three new bijections connecting various classes of lattice paths with growth-constrained sequences, expanding combinatorial understanding of these structures.

Contribution

It presents novel bijections between specific lattice path classes and growth-constrained integer sequences, enriching combinatorial bijective techniques.

Findings

Established a bijection between little Schr"{o}der paths and growth-constrained sequences.

Connected lattice paths with nonnegative slope steps to growth-constrained sequences.

Linked paths with steps (1,1) and (1,-j) to paths with steps (k,1) and (1,-1).

Abstract

We present three bijections, the first between little Schr\"{o}der paths and a class of growth-constrained integer sequences, the second between lattice paths consisting of steps with nonnegative slope and another class of growth-constrained sequences, the third between a class of lattice paths with steps $(1,1)$ and $(1,-j\,),\ j\ge 1$, and a class with steps $(k,1),\ k\ge 1$, and $(1,-1)$.