# Passing through a stack $k$ times

## TL;DR

This paper studies permutations sortable through multiple passes of a stack, characterizes classes based on the number of passes, and provides enumeration formulas and bounds for these classes.

## Contribution

It introduces the concept of $k$-pass sortable permutations, characterizes their classes, and establishes finite bases with bounds, along with a bijection to integer sequences for enumeration.

## Key findings

- All $k$-pass sortable classes have finite bases.
- Provides bounds on the basis element lengths for any $k$.
- Offers exact enumeration formulas for permutations based on tier and passes.

## Abstract

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation $\pi$ to be $k$-pass sortable if $\pi$ is sortable using $k$ passes through the stack. Permutations that are $1$-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of $2$-pass sortable permutations in terms of their basis. We also show all $k$-pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer $k$. Finally, we define the notion of tier of a permutation $\pi$ to be the minimum number of passes after the first pass required to sort $\pi$. We then give a bijection between the class of permutations of tier $t$ and a collection of integer sequences studied by Parker. This gives an exact enumeration of tier $t$ permutations of a given length and thus an exact enumeration for the class of $(t+1)$-pass sortable permutations. Finally, we give a new derivation for the generating function in Parker's thesis and an explicit formula for the coefficients.