Fertility monotonicity and average complexity of the stack-sorting map

Abstract

Let ${\mathcal{D}}_n$ denote the average number of iterations of West’s stack-sorting map $s$ that are needed to sort a permutation in $S_n$ into the identity permutation $123\cdots n$. We prove that

$$0.62433\approx \lambda \le \underset{n\to \infty }{lim\; inf}\frac{{\mathcal{D}}_n}{n}\le \underset{n\to \infty }{lim\; sup}\frac{{\mathcal{D}}_n}{n}\le \frac{3}{5}(7-8log2)\approx 0.87289,$$

where $\lambda$ is the Golomb–Dickman constant. Our lower bound improves upon West’s lower bound of 0.23, and our upper bound is the first improvement upon the trivial upper bound of 1. We then show that fertilities of permutations increase monotonically upon iterations of $s$. More precisely, we prove that $\left|s^{-1}\left(\sigma \right)\right|\le \left|s^{-1}\left(s\left(\sigma \right)\right)\right|$ for all $\sigma \in S_n$, where equality holds if and only if $\sigma =123\cdots n$. This is the first theorem that manifests a law-of-diminishing-returns philosophy for the stack-sorting map that Bóna has proposed. Along the way, we note some connections between the stack-sorting map and the right and left weak orders on $S_n$.

Introduction

Motivated by a problem involving sorting railroad cars, Knuth introduced a certain “stack-sorting” machine in his book The Art of Computer Programming [22]. Knuth’s analysis of this sorting machine led to several advances in combinatorics, including the notion of a permutation pattern and the kernel method [4], [6], [21], [24]. In his 1990 Ph.D. dissertation, West defined a deterministic variant of Knuth’s machine. This variant, which is a function that we denote by $s$, has now received a huge amount of attention (see [5], [6], [14], [15] and the references therein). West’s original definition makes use of a stack that is allowed to hold entries from a permutation. Here, a permutation is an ordering of a finite set of integers, written in one-line notation. Let $S_n$ denote the set of permutations of the set $\left[n\right]≔\{1,\dots ,n\}$. Assume we are given an input permutation $\pi ={\pi }_1\cdots {\pi }_n$. Throughout this procedure, if the next entry in the input permutation is smaller than the entry at the top of the stack or if the stack is empty, the next entry in the input permutation is placed at the top of the stack. Otherwise, the entry at the top of the stack is annexed to the end of the growing output permutation. This procedure stops when the output permutation has length $n$. We then define $s\left(\pi \right)$ to be this output permutation. Fig. 1 illustrates this procedure and shows that $s\left(4162\right)=1426$.

There is also a simple recursive definition of the map $s$. First, we declare that $s$ sends the empty permutation to itself. Given a nonempty permutation $\pi$, we can write $\pi =LmR$, where $m$ is the largest entry in $\pi$. We then define $s\left(\pi \right)=s\left(L\right)s\left(R\right)m$. For example,

$$s\left(5273614\right)=s\left(52\right)s\left(3614\right)7=s\left(2\right)5s\left(3\right)s\left(14\right)67=253s\left(1\right)467=2531467.$$

One of the central notions in the investigation of the stack-sorting map is that of a $t$-stack-sortable permutation, which is a permutation $\pi$ such that $s^t\left(\pi \right)$ is increasing ($s^t$ is the $t$-fold iterate of $s$). Let $W_t\left(n\right)$ be the number of $t$-stack-sortable permutations in $S_n$. The stack-sorting map moves the largest entry in a permutation to the end, so a simple inductive argument shows that every permutation of length $n$ is $(n-1)$-stack-sortable. It follows from Knuth’s analysis of his stack-sorting machine that the $1$-stack-sortable permutations are precisely the permutations that avoid the pattern $231$. Thus, $W_1\left(n\right)$ is the $n\text{th}$ Catalan number $C_n=\frac{1}{n+1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$. Settling a conjecture of West, Zeilberger [30] proved that $W_2\left(n\right)=\frac{2}{(n+1)(2n+1)}\left(\genfrac{}{}{0ex}{}{3n}{n}\right)$. The current author has obtained nontrivial asymptotic lower bounds for $W_t\left(n\right)$ for every fixed $t\ge 3$, and he has obtained nontrivial asymptotic upper bounds for $W_3\left(n\right)$ and $W_4\left(n\right)$ [13], [15]. He has also devised a polynomial-time algorithm for computing $W_3\left(n\right)$ [15]. Instead of focusing only on $t$-stack-sortable permutations when $t\ge 3$ is small and fixed, West realized that he could make progress if he attacked from the other side. He considered the cases $t=n-2$ and $t=n-3$. He showed that a permutation in $S_n$ is $(n-2)$-stack-sortable if and only if it does not end in the suffix $n1$ [29]. He also characterized and enumerated $(n-3)$-stack-sortable permutations in $S_n$. The case $t=n-4$ was treated in [8].

Define the stack-sorting tree on $S_n$ to be the rooted tree with vertex set $S_n$ in which the root is the identity permutation $123\cdots n$ and in which each nonidentity permutation $\pi$ is a child of $s\left(\pi \right)$. The stack-sorting depth of a permutation $\pi \in S_n$, which we denote by $ssd\left(\pi \right)$, is the depth of $\pi$ in this tree. Equivalently, $ssd\left(\pi \right)$ is the smallest nonnegative integer $t$ such that $\pi$ is $t$-stack-sortable. It is natural to view $s$ as a sorting algorithm that acts iteratively on an input permutation until reaching an increasing permutation. It requires $2n$ elementary operations to apply the map $s$ to a permutation in $S_n$, so $2nssd\left(\pi \right)$ is the time complexity of $s$ on the input $\pi$. We are interested in the quantity

$${\mathcal{D}}_n=\frac{1}{n!}\sum _{\pi \in S_n}ssd\left(\pi \right),$$

which is the average depth of the stack-sorting tree on $S_n$. Note that $2n{\mathcal{D}}_n$ is the average time complexity of the sorting algorithm that iteratively applies $s$. West [29] proved that

$$0.23\le \underset{n\to \infty }{lim\; inf}\frac{{\mathcal{D}}_n}{n}\le \underset{n\to \infty }{lim\; sup}\frac{{\mathcal{D}}_n}{n}\le 1,$$

where the upper bound of $1$ follows from the observation that $ssd\left(\pi \right)\le n-1$ for all $\pi \in S_n$. He also commented that it would probably not be possible to obtain a lower bound larger than $1∕2$ or an upper bound smaller than $1$ via his pattern-avoidance approach to the problem. Our first main result is as follows.

Theorem 1.1

We have

$$0.62433\approx \lambda \le \underset{n\to \infty }{lim\; inf}\frac{{\mathcal{D}}_n}{n}\le \underset{n\to \infty }{lim\; sup}\frac{{\mathcal{D}}_n}{n}\le \frac{3}{5}(7-8log2)\approx 0.87289,$$

where $\lambda$ is the Golomb–Dickman constant.

Another crucial notion in the study of the stack-sorting map is that of the fertility of a permutation $\pi$, which is simply $\left|s^{-1}\left(\pi \right)\right|$. Many problems concerning the stack-sorting map can be phrased in terms of fertilities. For example, computing $W_2\left(n\right)$ is equivalent to finding the sum of the fertilities of all of the $231$-avoiding (i.e., $1$-stack-sortable) permutations in $S_n$. The author found methods for computing fertilities of permutations [12], [13], [15], which led to the above-mentioned advancements in the investigation of $t$-stack-sortable permutations when $t\in \{3,4\}$. Permutations with fertility $1$ (called uniquely sorted permutations) possess some remarkable enumerative properties [14], [16], [25]. There is also a surprising connection between fertilities of permutations and a formula that converts from free to classical cumulants in noncommutative probability theory [11]; the author has used this connection to prove new results about the map $s$.

In Exercise 23 of Chapter 8 in [6], Bóna asks the reader to find the element of $S_n$ with the largest fertility. As one might expect, the answer is $123\cdots n$. The proof is not too difficult, but it is also not trivial. Our second main theorem generalizes this result by showing that the fertility statistic is strictly monotonically increasing as one moves up the stack-sorting tree.

Theorem 1.2

For every permutation $\sigma \in S_n$, we have

$$\left|s^{-1}\left(\sigma \right)\right|\le \left|s^{-1}\left(s\left(\sigma \right)\right)\right|,$$

where equality holds if and only if $\sigma =123\cdots n$.

Theorem 1.2 represents a step toward a law-of-diminishing-returns philosophy for the stack-sorting map that Miklós Bóna has postulated. Roughly speaking, his idea is that each successive iteration of the stack-sorting map should be less efficient in sorting permutations than the previous iterations. A concrete formulation of this idea manifests itself in Bóna’s conjecture that for each fixed $n\ge 1$, the sequence $W_1\left(n\right),W_2\left(n\right),\dots ,W_{n-1}\left(n\right)$ is log-concave (meaning $W_{t+1}\left(n\right)∕W_t\left(n\right)\ge W_{t+2}\left(n\right)∕W_{t+1}\left(n\right)$ for all $1\le t\le n-2$) [7]. Said differently, Bóna’s conjecture states that the average fertility of a $(t+1)$-stack-sortable permutation in $S_n$ is at most the average fertility of a $t$-stack-sortable permutation in $S_n$. While Theorem 1.2 does not imply this conjecture, it is a step in the right direction.

Remark 1.3

Suppose $\pi ={\pi }_1\cdots {\pi }_n\in S_n$, and let $i\in [n-1]$. If ${\pi }_i>{\pi }_{i+1}$, let $t_i\left(\pi \right)$ be the permutation obtained from $\pi$ by swapping the positions of the entries ${\pi }_i$ and ${\pi }_{i+1}$. If ${\pi }_i<{\pi }_{i+1}$, let $t_i\left(\pi \right)=\pi$. If $i+1$ appears to the left of $i$ in $\pi$, let ${\tilde{t}}_i\left(\pi \right)$ be the permutation obtained by swapping the positions of $i$ and $i+1$ in $\pi$. Otherwise, let ${\tilde{t}}_i\left(\pi \right)=\pi$. The right weak order on $S_n$ is the partial order ${\le }_{right}$ on $S_n$ defined by saying that ${\pi }^{\prime }{\le }_{right}\pi$ if there exists a sequence $i_1,\dots ,i_m$ of elements of $[n-1]$ such that $t_{i_m}\circ \cdots \circ t_{i_1}\left(\pi \right)={\pi }^{\prime }$. The left weak order on $S_n$ is the partial order ${\le }_{left}$ on $S_n$ defined by saying that ${\pi }^{\prime }{\le }_{left}\pi$ if there exists a sequence $i_1,\dots ,i_m$ of elements of $[n-1]$ such that ${\tilde{t}}_{i_m}\circ \cdots \circ {\tilde{t}}_{i_1}\left(\pi \right)={\pi }^{\prime }$.

Theorem 1.2 is a little bit strange in view of the relationship between the stack-sorting map and these two partial orders. It is not difficult to show that for every permutation $\sigma \in S_n$, we have $s\left(\sigma \right){\le }_{right}\sigma$. Therefore, one might expect to prove Theorem 1.2 by first establishing that $\left|s^{-1}\left({\pi }^{\prime }\right)\right|\ge \left|s^{-1}\left(\pi \right)\right|$ whenever ${\pi }^{\prime }{\le }_{right}\pi$. However, this turns out to be false. We have $31425{\le }_{right}34125$, but one can show that $\left|s^{-1}\left(31425\right)\right|=1<4=\left|s^{-1}\left(34125\right)\right|$. On the other hand, we will be able to prove (see Theorem 3.3) that

$$\left|s^{-1}\left({\pi }^{\prime }\right)\right|\ge \left|s^{-1}\left(\pi \right)\right|\text{whenever}{\pi }^{\prime }{\le }_{left}\pi .$$

Unfortunately, this inequality does not immediately imply Theorem 1.2 because the left weak order is not compatible with the action of the stack-sorting map. To see this, note that $s\left(231\right)=213⁄{\le }_{left}231$. Our proof of Theorem 1.2 will combine (1) with the Decomposition Lemma proved in [15]. $◊$