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Permutations sortable by two stacks in parallel and quarter plane walks

Michael Albert, Mireille Bousquet-Mélou

TL;DR

The paper addresses the long-standing problem of counting and characterizing permutations sortable by two parallel stacks. It develops a framework that translates stack-sorting into canonical operation sequences and quarter-plane lattice walks, yielding a pair of functional equations that define the generating function for sortable permutations and connect it to the generating function for quarter-plane walks with NW/ES corners. Central contributions include explicit relations S(t)=1+C(1−1/S, tS^2) and Q(−S•, t/(1+S•)^2)=(1+S•)/(1−S•), a detailed generating-function treatment of general and half-plane walks, and a set of conjectures on the core walk generating function Q(a,u) that drive the asymptotic analysis. Under these conjectures, the authors derive the asymptotic growth of sortable permutations with a predicted growth constant around 8.28 and an exponent near −2.5, linking permutation sorting to the rich theory of walks in cones and cone-like regions. The work opens avenues for rigorous proofs of the conjectures and suggests broader applicability to other “stack-like” rearranging devices and cone-walking problems.

Abstract

At the end of the 1960s, Knuth characterised the permutations that can be sorted using a stack in terms of forbidden patterns. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Even \& Itai, Pratt and Tarjan studied permutations that can be sorted using two stacks in parallel. This problem is significantly harder. In particular, a sortable permutation can now be sorted by several distinct sequences of stack operations. Moreover, in order to be sortable, a permutation must avoid infinitely many patterns. The associated counting question has remained open for 40 years. We solve it by giving a pair of functional equations that characterise the generating function of permutations that can be sorted with two parallel stacks. The first component of this system describes the generating function Q(a,u) of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with the series Q(a,u). We prove that they hold for loops confined to the upper half plane, or not confined at all. They remain open for quarter plane loops. Given the recent activity on walks confined to cones, we believe them to be attractive per se.

Permutations sortable by two stacks in parallel and quarter plane walks

TL;DR

The paper addresses the long-standing problem of counting and characterizing permutations sortable by two parallel stacks. It develops a framework that translates stack-sorting into canonical operation sequences and quarter-plane lattice walks, yielding a pair of functional equations that define the generating function for sortable permutations and connect it to the generating function for quarter-plane walks with NW/ES corners. Central contributions include explicit relations S(t)=1+C(1−1/S, tS^2) and Q(−S•, t/(1+S•)^2)=(1+S•)/(1−S•), a detailed generating-function treatment of general and half-plane walks, and a set of conjectures on the core walk generating function Q(a,u) that drive the asymptotic analysis. Under these conjectures, the authors derive the asymptotic growth of sortable permutations with a predicted growth constant around 8.28 and an exponent near −2.5, linking permutation sorting to the rich theory of walks in cones and cone-like regions. The work opens avenues for rigorous proofs of the conjectures and suggests broader applicability to other “stack-like” rearranging devices and cone-walking problems.

Abstract

At the end of the 1960s, Knuth characterised the permutations that can be sorted using a stack in terms of forbidden patterns. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Even \& Itai, Pratt and Tarjan studied permutations that can be sorted using two stacks in parallel. This problem is significantly harder. In particular, a sortable permutation can now be sorted by several distinct sequences of stack operations. Moreover, in order to be sortable, a permutation must avoid infinitely many patterns. The associated counting question has remained open for 40 years. We solve it by giving a pair of functional equations that characterise the generating function of permutations that can be sorted with two parallel stacks. The first component of this system describes the generating function Q(a,u) of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with the series Q(a,u). We prove that they hold for loops confined to the upper half plane, or not confined at all. They remain open for quarter plane loops. Given the recent activity on walks confined to cones, we believe them to be attractive per se.

Paper Structure

This paper contains 20 sections, 25 theorems, 145 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Canonical operation sequences
  3. Exact enumeration
  4. Corners in square lattice walks
  5. Some results on quarter plane loops
  6. General loops: Generating functions
  7. General loops: $\boldsymbol {(a+1)}$-positivity and radius of convergence
  8. Half plane walks
  9. Asymptotic Analysis
  10. Statement of the results
  11. Relating the singularities of $S$ and $C$
  12. Relating the singularities of $S$ and $Q$
  13. Some questions and observations
  14. Some questions raised directly by our work
  15. Other rearranging devices
  16. ...and 5 more sections

Key Result

Lemma 1

If a permutation can be produced by some operation sequence, then it can be produced by one that outputs eagerly.

Figures (12)

  • Figure 1: Four steps in the sequence of operations that outputs $2341$ from $1234$ using a stack. Each arrow shows an operation that is about to be performed.
  • Figure 2: The permutation $312$ cannot be produced with a single stack, but can be produced with two parallel stacks as shown here. Note that several distinct sequences of operations produce it.
  • Figure 3: An illustration of the arch system associated with the operation sequence $I_1I_2I_1I_1O_1O_1O_1I_1O_2O_1I_2I_1I_2O_2O_2O_1$, and its associated graph. The arches are labelled using the left-to-right order of their left endpoint. This arch system has five connected components, and one left-right pair (between arches 2 and 5). The output permutation is $43125867$.
  • Figure 4: The canonical arch system that is equivalent to the arch system of Figure \ref{['fig:ex']}. Note that the left-right pair created by edges 2 and 5 in Figure \ref{['fig:ex']} has disappeared (these edges do not cross any more). Also, the colours of the two rightmost components (edges 6, 7, 8) have changed. The output permutation is still $43125867$.
  • Figure 5: The structure of an arch system: a connected system $c$ with $n$ arches (here, $n=3$), in which $2n$ arbitrary arch systems are inserted. Here, $c$ has two left-right pairs. The arch systems that are inserted there (shown in white) destroy these left-right pairs, unless they are empty.
  • ...and 7 more figures

Theorems & Definitions (54)

  • Lemma 1
  • proof
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • ...and 44 more