Continuously Increasing Subsequences of Random Multiset Permutations

Given integers $m$ and $n$, let $\mathfrak{S}_{m,n}$ denote the set of words $\pi$ which use each integer in $[n]:=\{1,2,\ldots,n\}$ exactly $m$ times, and we will refer to $\pi\in\mathfrak{S}_{m,n}$ as a multiset permutation. For example, $\pi=211323\in\mathfrak{S}_{2,3}$. For $\pi\in\mathfrak{S}_{m,n}$ and $i$ an integer, we define $L^{i}_{m,n}(\pi)$ to be the length of the longest subsequence of $\pi$ of the form $i(i+1)(i+2)\cdots j$, which we call an $i$-continuously increasing subsequence. We say that a subsequence is a continuously increasing subsequence if it is an $i$-continuously increasing subsequence for some $i$, and we define $L_{m,n}(\pi)=\max_{i}L^{i}_{m,n}(\pi)$ to be the length of a longest continuously increasing subsesquence of $\pi$. For example, if $\pi=2341524315$ then $L_{2,5}(\pi)=L^{2}_{2,5}(\pi)=4$ due to the subsequence $2345$, and $L^{1}_{2,5}(\pi)=3$ due to the subsequence $123$.

The focus of this paper is to study $L^{1}_{m,n}(\pi)$ and $L_{m,n}(\pi)$ when $\pi$ is chosen uniformly at random from $\mathfrak{S}_{m,n}$. We focus on the regime where $n$ is much larger than $m$, as in the opposite regime $L^{i}_{m,n}(\pi)$ is very likely to be its maximum possible value $n-i+1$ for all $i$.

We first consider $\mathbb{E}[L^{1}_{m,n}(\pi)]$. This quantity was briefly studied by Diaconis, Graham, He, and Spiro [DGHS21] due to its relationship with a certain card game that we describe later in this paper. They proved $\mathbb{E}[L^{1}_{m,n}(\pi)]\leq m+Cm^{3/4}\log m$ for some absolute constant $C$ provided $n$ is sufficiently large in terms of $m$. It was conjectured in [DGHS21] that this upper bound for $\mathbb{E}[L^{1}_{m,n}(\pi)]$ is asymptotically tight for $n$ sufficiently large in terms of $m$. We verify this conjecture in a strong form, obtaining an exact formula for $\lim_{n\to\infty}\mathbb{E}[L^{1}_{m,n}(\pi)]$ for any fixed $m$ and precise estimates of this value as $m$ tends towards infinity.

Note when $m=1$ the error term of Theorem 1.3 is $\Theta(1)$, but for $m\geq 2$ it is $\Theta(\frac{\log m}{\log\Gamma^{-1}(n)}\Gamma^{-1}(n))$, which is fairly close to the main term of $\Gamma^{-1}(n)$. Thus the behavior of $\mathbb{E}[L_{m,n}(\pi)]$ changes somewhat dramatically as soon as one starts to consider multiset permutations as opposed to just permutations.

Determining $L^{i}_{m,n}(\pi)$ and $L_{m,n}(\pi)$ can be viewed as variants of the well-studied problem of determining the length of the longest increasing subsequence in a random permutation of length $n$, and we denote this quantity by $\widetilde{L}_{n}$. It was shown by Logan and Shepp [LS77] and Vershick and Kerov [VK77] that $\mathbb{E}[\widetilde{L}_{n}]\sim 2\sqrt{n}$, answering a famous problem of Ulam. Later Baik, Deift, and Johansson [BDJ99] showed that the limiting distribution of $\widetilde{L}_{n}$ is the Tracy-Widom distribution. Some work with the analogous problem for multiset permutations has been considered recently by Almeanazel and Johnson [AMJ20]. Much more can be said about this topic, and we refer the reader to the excellent book by Romik [Rom15] for more information.

The initial motivation for studying $L^{1}(\pi)$ was due to its relationship to a card guessing experiment introduced by Diaconis and Graham [DG81]. To start the experiment, one shuffles a deck of $mn$ cards which consists of $n$ distinct card types each appearing with multiplicity $m$. In each round, a subject iteratively guesses what the top card of the deck is according to some strategy $G$. After each guess, the subject is told whether their guess was correct or not, the top card is discarded, and then the experiment continues with the next card. This experiment is known as the partial feedback model. For more on the history of the partial feedback model we refer the reader to [DGS21].

If $G$ is a strategy for the subject in the partial feedback model and $\pi\in\mathfrak{S}_{m,n}$, we let $P(G,\pi)$ denote the number of correct guesses made by the subject if they follow strategy $G$ and the deck is shuffled according to $\pi$. We say that $G$ is an optimal strategy if $\mathbb{E}[P(G,\pi)]=\max_{G^{\prime}}\mathbb{E}[P(G^{\prime},\pi)]$, where $G^{\prime}$ ranges over all strategies and $\pi\in\mathfrak{S}_{m,n}$ is chosen uniformly at random. Optimal strategies are unknown in general, and even if they were known they would likely be too complex for a human subject to implement in practice. As such there is interest in coming up with (simple) strategies $G$ such that $\mathbb{E}[P(G,\pi)]$ is relatively large.

One strategy is the trivial strategy which guesses card type 1 every single round, guaranteeing a score of exactly $m$ at the end of the experiment. A slightly better strategy is the safe strategy $G_{safe}$ which guesses card type 1 every round until all $m$ are guessed correctly, then 2’s until all $m$ are guessed correctly, and so on. It can be deduced from arguments given by Diaconis, Graham, and Spiro [DGS21] that $\mathbb{E}[P(G_{safe},\pi)]$ is $m+1-\frac{1}{m+1}$ plus an exponential error term, so the safe strategy does just a little better than the trivial strategy.

Another natural strategy is the shifting strategy $G_{shift}$, defined by guessing 1 until you are correct, then 2 until you are correct, and so on; with the strategy being defined arbitrarily in the (very rare) event that one correctly guesses a copy of each card type. It is not difficult to see that $P(G_{shift},\pi)\geq L^{1}_{m,n}(\pi)$, with equality holding provided the player does not correctly guess $n$. Thus Theorem 1.1(b) shows that the expected number of correct guesses under the shifting strategy is close to $m+1-\frac{1}{m+2}$, which is slightly better than the trivial strategy, and very slightly better than the safe strategy.

We let $[n]:=\{1,2,\ldots,n\}$ and let $[m]^{n}$ be the set of tuples of length $n$ with entries in $[m]$. Whenever we write, for example, $\Pr[L_{m,n}(\pi)\geq k]$, we will assume $\pi$ is chosen uniformly at random from $\mathfrak{S}_{m,n}$ unless stated otherwise.

Notice that $\mathcal{L}_{m}^{1}$ can be expressed in terms of $h_{m}(n)$ as follows:

where the third equality is due to Observation 1.4. Note that $|\mathfrak{S}_{m,n}|=(mn)!/(m!)^{n}$. Thus, as a consequence of Theorem 2.1, we have

Intuitively, if the $1/l!$ were removed from the right-hand-side expression in (3b), then by using the multinomial theorem we could write this expression as an $n^{\text{th}}$ power, turning (2) into a geometric series. The next few paragraphs formalise this idea.

We begin by replacing $(-1)^{l}$ by $x^{l}$ in the right-hand-side of (3a) to obtain the polynomial

Thus,

Next, we define an operator in order to remove the $l!$ from the denominator. Let $R$ be a commutative ring containing $\mathbb{Q}$ and let $\Phi:R[x]\to R[x]$ be an $R$-linear map defined on the monomials by

We can extend $\Phi$ to an $R$-linear map on $R[[x]]\to R[[x]]$, which we also refer to as $\Phi$ by abuse of notation. Throughout this article, $R$ is either $\mathbb{C}$ or $\mathbb{C}[[y]]$ for an indeterminate $y$, and we shall refer to this $R$-linear map as $\Phi$ in both cases. Notice that $\Phi$ is invertible for any such ring $R$. A key property that we use about $\Phi$ is

Consider the polynomial

Notice that,

Let $P_{m}(x,y)$ and $Q_{m}(x,y)$ be the ordinary generating functions of $p_{m,n}(x)$ and $q_{m,n}(x)$ respectively, i.e.

Putting everything together, we see that

and thus it suffices to find a nice closed form expression for $P_{m}(x,y)$. Note that

where we recall the polynomial $E_{m-1}(x)$ is defined in Theorem 1.1 by $E_{m-1}(x)=\sum_{k=0}^{m-1}x^{k}/k!$. As $q_{m,n}(x)=\left(q_{m,1}(x)\right)^{n}$, we have

Hence,

and thus

We now prove the main result of this subsection.

This proposition together with (7) gives Theorem 1.1(a).

We remind the reader that Theorem 1.1(b) claims

for some constant $\beta>0$. Given Theorem 1.1(a), proving this claim is equivalent to showing that $\displaystyle\sum_{i}\alpha_{i}^{-1}e^{-\alpha_{i}}=-m-2+\frac{1}{m+2}+O(e^{-\beta m})$ for some positive constant $\beta$. We do this by following the approach used by Conrey and Ghosh [CG88] to evaluate a similar exponential sum.

For $1\leq i\leq m$, let $s_{i}$ and $t_{i}$ be the real and imaginary parts, respectively, of $\alpha_{i}$. Let $\gamma^{-},\gamma,\gamma^{+}$ be arbitrary positive numbers such that $0<\gamma^{-}<\gamma<\gamma^{+}<1-\log{2}$. We partition $[m]$ into disjoint sets $S$ and $L$ where $i\in S$ when $s_{i}\leq{\gamma m}$ and $i\in L$ when $s_{i}>\gamma m$, allowing us to rewrite our desired sum as

To evaluate the sum for the indices in $S$, we utilize $R_{m}(x)$. Since the $\alpha_{i}$’s are the roots of $E_{m}(x)$, we have $e^{\alpha_{i}}=R_{m}(\alpha_{i})$ for $i=1,\cdots,m$, so $e^{-\alpha_{i}}=\frac{1}{R_{m}(\alpha_{i})}$.

Note that for $i\in S$ we have $|\alpha_{i}|<m/2$ from Proposition 2.3(a). Thus we can use Lemma 2.5 to conclude that

where $c_{0}=1$ and $c_{1}=\frac{-1}{m+2}$.

Using the values of $\displaystyle\sum_{i=1}^{m}\alpha_{i}^{-t}$ for $t=1,\cdots,m+2$ from Proposition 2.3(c) and that $c_{0}=1$ and $c_{1}=\frac{-1}{m+2}$, we see that

Plugging this into (9) gives

By Lemma 2.4, we have $\displaystyle\sum_{i=1}^{m}\alpha_{i}^{-1}e^{-\alpha_{i}}=\displaystyle\sum_{i\in S}\alpha_{i}^{-1}e^{-\alpha_{i}}+O(e^{-\beta m})$ for some $\beta>0$, so we are left to consider the sum over $S$. The first three terms above match our claimed expression, so it suffices to show that the leftover terms $-(m+1)!\left(c_{m+1}+\displaystyle\sum_{k=0}^{m+1}\displaystyle\sum_{i\in L}c_{k}\alpha_{i}^{k-m-2}-\displaystyle\sum_{i\in S}\displaystyle\sum_{k=m+2}^{\infty}c_{k}\alpha_{i}^{k-m-2}\right)$ are $O(e^{-\beta m})$.

Using the Triangle Inequality, and recalling that $|c_{k}|\leq{(\frac{2}{m})^{k}}$ for all $k$ by Proposition 2.3(b), we obtain