On the Generalized Climbing Stairs Problem

TL;DR

This paper develops a generating function approach to count the ways to climb stairs with variable step sizes and multiplicities, linking to integer sequences and proposing open questions on compositions.

Contribution

It introduces a generating function solution for counting stair-climbing methods with constraints, connecting to OEIS sequences and exploring open problems.

Findings

Derived explicit generating function for constrained stair-climbing counts

Linked stair-counting problem to ten OEIS sequences

Proposed open questions on counting compositions of n

Abstract

Let $\mathcal S$ be a subset of the positive integers, and $M$ be a positive integer. Mohammad K. Azarian, inspired by work of Tony Colledge, considered the number of ways to climb a staircase containing $n$ stairs using "step-sizes" $s \in \mathcal S$ and multiplicities at most $M$. In this exposition, we find a solution via generating functions, i.e., an expression which counts the number of partitions $n = \sum_{s \in \mathcal S} m_s s$ satisfying $0 \leq m_s \leq M$. We then use this result to answer a series of questions posed by Azarian, thereby showing a link with ten sequences listed in the On-Line Encyclopedia of Integer Sequences. We conclude by posing open questions which seek to count the number of compositions of $n$.