# Patterns in Random Permutations Avoiding Some Sets of Multiple Patterns

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## Abstract

We consider a random permutation drawn from the set of permutations of length *n* that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern \(\sigma \) has a limit distribution, after suitable scaling. In several cases, the number is asymptotically normal; this contrasts to the cases of permutations avoiding a single pattern of length 3 studied in earlier papers.

## Keywords

Random permutations Patterns in permutations Forbidden patterns## Mathematics Subject Classification

60C05 05A05 05A16 60F05## 1 Introduction

*occurrence*of \(\sigma \) in \(\pi \) is a subsequence \(\pi _{i_1}\cdots \pi _{i_m}\), with \(1\leqslant i_1<\dots <i_m\leqslant n\), that has the same order as \(\sigma \), i.e., \(\pi _{i_j}<\pi _{i_k} \iff \sigma _j<\sigma _k\) for all \(j,k\in [m]\). We let \(n_\sigma (\pi )\) be the number of occurrences of \(\sigma \) in \(\pi \), and note that

*avoids*another permutation \(\tau \) if \(n_\tau (\pi )=0\); otherwise, \(\pi \)

*contains*\(\tau \). Let

*n*that avoid \(\tau \). More generally, for any set \(T=\{\tau _1,\dots ,\tau _k\}\) of permutations, let

*n*that avoid all \(\tau _i\in T\). We also let \(\mathfrak {S}_*(T):=\bigcup _{n=1}^\infty \mathfrak {S}_n(T)\) be the set of

*T*-avoiding permutations of arbitrary length.

The classes \(\mathfrak {S}_*(\tau )\) and, more generally, \(\mathfrak {S}_*(T)\) have been studied for a long time, see e.g. Knuth [16, Exercise 2.2.1–5], Simion and Schmidt [20], Bóna [3]. In particular, one classical problem is to enumerate the sets \(\mathfrak {S}_n(\tau )\), either exactly or asymptotically, see Bóna [3, Chapters 4–5]. We note the fact that for any \(\tau \) with length \(|\tau |=3\), \(\mathfrak {S}_n(\tau )\) has the same size \( |\mathfrak {S}_n(\tau )|=C_n:=\frac{1}{n+1}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \), the *n*th Catalan number, see e.g. [16, Exercises 2.2.1–4,5], [20, 21, Exercise 6.19ee,ff], [3, Corollary 4.7]; furthermore, the cases when *T* consists of several permutations of length 3 are all treated by Simion and Schmidt [20]. (The situation for \(|\tau |\geqslant 4\) is more complicated.)

The general problem that concerns us is to take a fixed set *T* of one or several permutations and let \(\varvec{\pi }_{{T};n}\) be a uniformly random *T*-avoiding permutation, i.e., a uniformly random element of \(\mathfrak {S}_n(T)\), and then study the distribution of the random variable \(n_\sigma (\varvec{\pi }_{{T};n})\) for some other fixed permutation \(\sigma \). (Only \(\sigma \) that are themselves *T*-avoiding are interesting, since otherwise \(n_\sigma (\varvec{\pi }_{{T};n})=0\).) One instance of this problem was studied already by Robertson, Wilf and Zeilberger [18], who gave a generating function for \(n_{123}(\varvec{\pi }_{{132};n})\). The exact distribution of \(n_\sigma (\varvec{\pi }_{{\tau };n})\) for a given *n* was studied numerically in [15], where higher moments and mixed moments are calculated for small *n*. We are mainly interested in asymptotics of the distribution of \(n_\sigma (\varvec{\pi }_{{T};n})\), and of its moments, as \({n\rightarrow \infty }\), for some fixed *T* and \(\sigma \).

In the present paper we study the cases when *T* is a set of two or more permutations of length 3. We consider 8 different cases separately; by symmetries, see Sect. 2.2, these cover all 30 non-trivial cases of such forbidden sets *T*. (See Sect. 2.5 for the trivial cases.) These 8 cases are included (together with a few others for reference) in Table 1.

The cases when \(T=\{\tau \}\) for a single permutation \(\tau \) of length \(|\tau |=3\) were studied in [12, 13] (by symmetries, only two such cases have to be considered), and the cases when *T* contains a permutation of length \(\leqslant 2\) are trivial (there is then at most one permutation in \(\mathfrak {S}_n(T)\) for any *n*); hence the present paper completes the study of forbidding one or several permutations of length \(\leqslant 3\). The case of forbidding one or several permutations of length \(\geqslant 4\) seems much more complicated, but there are recent impressive results in some cases by Bassino et al. [2] and Bassino et al. [1].

*T*-avoiding permutations, has previously been treated in a number of papers for various cases, beginning with Bóna [5, 7] (with \(\tau ={132}\)). In particular, Zhao [23] has given exact formulas when \(|\sigma |=3\) for the (non-trivial) cases treated in the present paper, where

*T*consists of two or more permutations of length 3.

The table shows whether \(n_\sigma (\varvec{\pi }_{{T};n})\) has limits of type I (normal) or II (non-normal); furthermore, the exponent \(\alpha =\alpha (\sigma )\) such that \({\mathbb {E}{}}n_\sigma (\varvec{\pi }_{{T};n})\) is of order \(n^\alpha \) is given in the column for the type. The last column shows the exceptional cases, if any, where the asymptotic variance vanishes. \(C_n:=\frac{1}{n+1}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \) is a Catalan number; \(F_{n+1}\) is a Fibonacci number (\(F_0=0\), \(F_1=1\)); \(s_{n-1}\) is a Schröder number; \(D(\sigma )\) is the number of descents and \(B(\sigma )\) is the number of blocks in \(\sigma \)

| \(|\mathfrak {S}_n(T)|\) | Type I | Type II | As. variance = 0 |
---|---|---|---|---|

\(\emptyset \) |
| \(|\sigma |\) | ||

\(\{132\}\) | \(C_n\) | \((|\sigma |+D(\sigma ))/2\) | \(m\cdots 1\) | |

\(\{321\}\) | \(C_n\) | \((|\sigma |+B(\sigma ))/2\) | \(1\cdots m\) | |

\(\{{132}, {312}\}\) | \(2^{n-1}\) | \(|\sigma |\) | ||

\(\{{231}, {312}\}\) | \(2^{n-1}\) | \(B(\sigma )\) | \(1\cdots m\) | |

\(\{{231}, {321}\}\) | \(2^{n-1}\) | \(B(\sigma )\) | \(1\cdots m\) | |

\(\{{132}, {321}\}\) | \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) +1\) | \(|\sigma |\) | ||

\(\{{231},{312}, {321}\}\) | \(F_{n+1}\) | \(B(\sigma )\) | \(1\cdots m\) | |

\(\{{132},{231}, {312}\}\) | | \(|\sigma |\) | ||

\(\{{132},{231}, {321}\}\) | | \(|\sigma |-1\) or \(|\sigma |\) | \(1\cdots m\) | |

\(\{{132},{213}, {321}\}\) | | \(|\sigma |\) | ||

\(\{{2413,3142}\}\) | \(s_{n-1}\) | \(|\sigma |\) |

The cases studied here (except some trivial, degenerate cases), all have asymptotic distributions of one of the following two types. We denote convergence in distribution by \(\overset{\mathrm {d}}{\longrightarrow }\) and convergence in probability by \(\overset{\mathrm {p}}{\longrightarrow }\).

**Normal Limits.**For the non-restricted case of uniformly random permutations in \(\mathfrak {S}_n\), it is well-known that if \(\varvec{\pi }_n\) is a uniformly random permutation in \(\mathfrak {S}_n\), then \(n_\sigma (\varvec{\pi }_n)\) has an asymptotic normal distribution as \({n\rightarrow \infty }\) for every fixed permutation \(\sigma \); more precisely, if \(|\sigma |=m\geqslant 2\) then, as \({n\rightarrow \infty }\),

**Non-normal Limits Without Concentration.**On the other hand, in other cases (Sects. 3, 7, 9, 10, 11) we find a different type of limit, where

*T*). The same holds in the case \(T=\{2413,3142\}\) studied by Bassino, Bouvel, Féray, Gerin and Pierrot [2].

We summarize the results of the present paper, together with some older results from [2, 4, 6, 12, 13, 15], in Table 1; for reference, we include the number \(|\mathfrak {S}_n(T)|\) of *T*-avoiding permutations of length *n*, see e.g. [16, Exercises 2.2.1-4,5], [21, Exercise 6.19ee,ff], [3, Corollary 4.7], [20], and [22]. We see no obvious pattern in the occurence of the two types of limits; nor do we know whether these are the only possibilities for a general set *T* of forbidden permutations.

### Remark 1.1

In the present paper we consider for simplicity often only univariate limits; corresponding multivariate results for several \(\sigma _1,\dots ,\sigma _k\) follow by the same methods. In particular, (1.4) and all instances of normal limit laws below extend to multivariate normal limits, with covariance matrices that can be computed explicitly.

### Remark 1.2

In the present paper we study only the numbers \(n_\sigma \) of occurences of some pattern in \(\varvec{\pi }_{{\tau };n}\). There is also a number of papers by various authors that study other properties of random \(\tau \)-avoiding permutations, see e.g. the references in [13]; such results will not be considered here.

## 2 Preliminaries

### 2.1 Notation

Let \(\iota =\iota _n\) be the identity permutation of length *n*. Let \(\bar{\iota }_n=n\cdots 21\) be its reversal.

Let \(\pi =\pi _1\cdots \pi _n\) be a permutation. We say that a value \(\pi _i\) is a *maximum* if \(\pi _i>\pi _j\) for every \(j<i\), and a *minimum* if \(\pi _i<\pi _j\) for every \(j<i\). (These are sometimes called *LR maximum* and *LR minimum*.) Note that \(\pi _1\) always is both a maximum and a minimum.

By joint convergence in distribution for an infinite family of random variables (depending on *n*) we mean convergence in the product topology, i.e., joint convergence for every finite subset.

### 2.2 Symmetries

*inverse*\(\pi ^{-1}\) in the usual way, and its

*reversal*and

*complement*by

*T*of permutations we define \(T^{\mathsf s}:=\{\tau ^{\mathsf s}:\tau \in T\}\). It follows from (2.4) that

*T*and \(T^{\mathsf s}\) are

*equivalent*, and note that (2.6) implies that it suffices to consider one set

*T*in each equivalence class \(\{T^{\mathsf s}:{\mathsf s}\in \mathfrak G\}\). We do this in the sequel without further comment. (We choose representatives

*T*that we find convenient. One guide is that we choose

*T*such that the identity permutation \(\iota _n\) avoids

*T*.)

### 2.3 Compositions and Decompositions of Permutations

If \(\sigma \in \mathfrak {S}_m\) and \(\tau \in \mathfrak {S}_n\), their *composition*\(\sigma *\tau \in \mathfrak {S}_{m+n}\) (in the literature often denoted \(\sigma \oplus \tau \)) is defined by letting \(\tau \) act on \([m+1,m+n]\) in the natural way; more formally, \(\sigma *\tau =\pi \in \mathfrak {S}_{m+n}\) where \(\pi _i=\sigma _i\) for \(1\leqslant i\leqslant m\), and \(\pi _{j+m}=\tau _j+m\) for \(1\leqslant j\leqslant n\). It is easily seen that \(*\) is an associative operation that makes \(\mathfrak {S}_*\) into a semigroup (without unit, since we only consider permutations of length \(\geqslant 1\)). We say that a permutation \(\pi \in \mathfrak {S}_*\) is *decomposable* if \(\pi =\sigma *\tau \) for some \(\sigma ,\tau \in \mathfrak {S}_*\), and *indecomposable* otherwise; we also call an indecomposable permutation a *block*. Equivalently, \(\pi \in \mathfrak {S}_n\) is decomposable if and only if \(\pi :[m]\rightarrow [m]\) for some \(1\leqslant m<n\). See e.g. [8, Exercise VI.14].

It is easy to see that any permutation \(\pi \in \mathfrak {S}_*\) has a unique decomposition \(\pi =\pi _1*\dots *\pi _\ell \) into indecomposable permutations (blocks) \(\pi _1,\dots ,\pi _\ell \) (for some, unique, \(\ell \geqslant 1\)); we call these the *blocks of*\(\pi \).

We shall see that some (but not all) of the classes considered below can be characterized in terms of their blocks. (See [2] for another, more complicated, example.)

### 2.4 \(U\)-statistics

*f*is a given function of \(d\geqslant 1\) variables. These were (in the symmetric case) introduced by Hoeffding [9]; see further e.g. [14] and the references there. We say that

*d*is the

*order*of the \(U\)-statistic.

*X*denoting a generic \(X_i\),

### Proposition 2.1

Moreover, if \(f(X_1,\dots ,X_d)\in L^p\) for some \(p\geqslant 2\), the (2.12) holds with convergence of all moments of order \(\leqslant p\). \(\square \)

### Example 2.2

A uniformly random permutation \(\varvec{\pi }_n\) of length *n* (without other restrictions) can be constructed as the relative order of \(X_1,\dots ,X_n\), where \(X_i\) are i.i.d. with, for example, a uniform distribution \(\mathsf U(0,1)\). For any given permutation \(\sigma \in \mathfrak {S}_m\), we can then write \(n_\sigma (\varvec{\pi }_n)\) as a \(U\)-statistic (2.7) for a suitable indicator function *f*. Then Proposition 2.1 yields a limit theorem showing that \(n_\sigma (\varvec{\pi }_n)\) is asymptotically normal. See [15] for details.

### Remark 2.3

The definitions (2.13)–(2.14) differ slightly from the ones in [14], where instead \(S_n\leqslant x\) and \(S_n>x\) are used. This does not affect the asymptotic results used here. Note that the event \(\{S_k=n \text { for some }k\geqslant 0\}\) equals \(\{S_{N_{+}(n)}=n\}\) in the present notation.

The following results are special cases of [14, Theorems 3.11, 3.13(iii) and 3.18] (with somewhat different notation). \(N_{\pm }(x)\) means either \(N_{-}(x)\) or \(N_{+}(x)\); the results holds for both.

### Proposition 2.4

### 2.5 Trivial Cases

- (i)
\(T=\{123,321\}\),

- (ii)
\(|T|=3\) and \(T\supset \{123,321\}\),

- (iii)
\(|T|\geqslant 4\).

## 3 Avoiding a Single Permutation of Length 3

There are 6 cases where a single permutation of length 3 is avoided, but by the symmetries in Sect. 2.2 these reduce to 2 non-equivalent cases, for example 132 (equivalent to 231, 213, 312) and 321 (equivalent to 123). These cases are treated in detail in [12] and [13], respectively. Both analyses are based on bijections with binary trees and Dyck paths, and the well-known convergence in distribution of random Dyck paths to a Brownian excursion, but the details are very different, and so are in general the resulting limit distributions.

For comparison with the results in later sections, we quote the main results of [12] and [13], referring to these papers for further details and proofs. Recall that the standard Brownian excursion \(\mathbf {e}(x)\) is a random non-negative function on \([0,1]\).

*descents*in \(\sigma \), i.e., indices

*i*such that \(\sigma _i>\sigma _{i+1}\) or (as a convenient convention) \(i=|\sigma |\). Note that \(1\leqslant D(\sigma )\leqslant |\sigma |\), and thus

### Theorem 3.1

For a monotone decreasing permutation \(k\cdots 1\), \(\Lambda _{k\cdots 1}=1/k!\) is deterministic, but not for any other \(\sigma \). \(\square \)

The limit variables \(\Lambda _\sigma \) in Theorem 3.1 can be expressed as functionals of a Brownian excursion \(\mathbf {e}(x)\), see [12]; the description is, in general, rather complicated, but some cases are simple.

### Example 3.2

In the special case \(\sigma =12\), \(\Lambda _{12}=\sqrt{2}\int _0^1\mathbf {e}(x)\,\mathrm {d}x\), see [12, Example 7.6]; this is (apart from the factor \(\sqrt{2}\)) the well-known *Brownian excursion area*, see e.g. [11] and the references there.

### Theorem 3.3

Moreover, the convergence (3.5) holds jointly for any set of \(\sigma \in \mathfrak {S}_*({321})\), and with convergence of all moments.

### Example 3.4

## 4 Avoiding \(\{{132}, {312}\}\)

In this section we avoid \(T=\{{132}, {312}\}\). Equivalent sets are \(\{132,231\}\), \(\{213, 231\}\), \(\{213, 312\}\).

It was shown by Simion and Schmidt [20] that \(|\mathfrak {S}_n({132}, {312})|=2^{n-1}\), together with the following characterization (in an equivalent formulation).

### Proposition 4.1

[20, Proposition 12] A permutation \(\pi \) belongs to the class \(\mathfrak {S}_*({132}, {312})\) if and only if every entry \(\pi _i\) is either a maximum or a minimum. \(\square \)

We encode a permutation \(\pi \in \mathfrak {S}_n({132}, {312})\) by a sequence \(\xi _2,\dots ,\xi _n\in \{\pm 1\}^{n-1}\), where \(\xi _j=1\) if \(\pi _j\) is a maximum in \(\pi \), and \(\xi _j=-1\) if \(\pi _j\) is a minimum. This is by Proposition 4.1 (and its proof in [20]) a bijection, and hence the code for a uniformly random \(\varvec{\pi }_{{{132}, {312}};n}\) has \(\xi _2,\dots ,\xi _n\) i.i.d. with the symmetric Bernoulli distribution \({\mathbb {P}{}}(\xi _j=1)={\mathbb {P}{}}(\xi _j=-1)=\frac{1}{2}\). We let \(\xi _1\) have the same distribution and be independent of \(\xi _2,\dots ,\xi _n\).

*f*does not depend on the first argument. It follows that, with the notation (2.8)–(2.11),

### Theorem 4.2

Moreover, (4.7) holds with convergence of all moments.

### Example 4.3

### Remark 4.4

It is easily seen from (4.1)–(4.2) that the expected number of occurrences \({\mathbb {E}{}}n_\sigma (\varvec{\pi }_{{{132}, {312}};n}) =2^{1-m}\left( {\begin{array}{c}n\\ m\end{array}}\right) \), for every \(\sigma \in \mathfrak {S}_m({132}, {312})\); hence the expectation depends only on the length \(m=|\sigma |\).

The variance depends not only on \(|\sigma |\), not even asymptotically, by (4.6).

## 5 Avoiding \(\{{231}, {312}\}\)

In this section we consider \(T=\{{231}, {312}\}\). The only equivalent set is \(\{132, 213\}\).

It was shown by Simion and Schmidt [20] that \(|\mathfrak {S}_n({231}, {312})|=2^{n-1}\), together with the following characterization (in an equivalent form).

### Proposition 5.1

[20, Proposition 12] A permutation \(\pi \) belongs to the class \(\mathfrak {S}_*({231}, {312})\) if and only if every block in \(\pi \) is decreasing, i.e., of the type \(\ell (\ell -1)\cdots 21\) for some \(\ell \). \(\square \)

*B*is random. However, we can analyze this variable using the renewal theory in Sect. 2.4 as follows.

First, mark each endpoint of the blocks in \(\pi \in \mathfrak {S}_n({231}, {312})\) by 1, and mark all other indices in [*n*] by 0. Thus \(\pi \) defines a string \(\xi _1,\dots ,\xi _n\in \{0,1\}^n\), where necessarily \(\xi _n=1\) but \(\xi _1,\dots ,\xi _{n-1}\) are arbitrary. This yields a bijection between \(\mathfrak {S}_n({231}, {312})\) and the \(2^{n-1}\) such strings; hence, we obtain a uniformly random \(\varvec{\pi }_{{{231}, {312}};n}\) by letting \(\xi _1,\dots ,\xi _{n-1}\) be i.i.d. \(\mathrm{Be}(\frac{1}{2})\), i.e., with \({\mathbb {P}{}}(\xi _i=0)={\mathbb {P}{}}(\xi _i=1)=\frac{1}{2}\).

*i*with \(\xi '_i=1\) as the end of a block, and let \(X_1,X_2,\dots \), be the successive lengths of these (infinitely many) blocks. Then \(X_i\) are i.i.d. with

*n*, we then may let \(\xi _i:=\xi '_i\) for \(1\leqslant i<n\), and \(\xi _n:=1\); this determines \(\xi _1,\dots ,\xi _n\) as above, and thus a uniformly random \(\varvec{\pi }_{{{231}, {312}};n}\). With this construction, the number of blocks in \(\varvec{\pi }_{{{231}, {312}};n}\) is, recalling (2.13)–(2.14), \(B=N_{+}(n)\), and the block lengths are

### Remark 5.2

Alternatively, we can obtain \((\xi _i)\) from \((\xi '_i)\) by conditioning on \(\xi '_n=1\), and note that this holds when \(S_{N_{+}(n)}=n\) (see Remark 2.3), and then \(n_\sigma (\varvec{\pi }_{{{231}, {312}};n}) = U_{N_{+}(n)}\). The result then follows from Proposition 2.5.

*X*has the probability generating function

### Theorem 5.3

Moreover, (5.19) holds with convergence of all moments. \(\square \)

### Example 5.4

### Remark 5.5

Theorem 5.3 shows that the typical order of \(n_\sigma (\varvec{\pi }_{{{231}, {312}};n})\) depends only on the number of blocks *b* in \(\sigma \) (but not on the length \(|\sigma |\)); more precisely, the asymptotic mean depends only on *b*. (Cf. the different situation when avoiding \(\{{132}, {312}\}\) in Sect. 4, see Remark 4.4.) Calculations (assisted by Maple) show, however, that the asymptotic variance \(\gamma ^2\) depends not only on *m* and *b*; for example \(\sigma =2143=\bar{\iota }_2*\bar{\iota }_2\) has \(\gamma ^2=6\) while \(\sigma =3214=\bar{\iota }_3*\bar{\iota }_1\) has \(\gamma ^2=52/3\).

### Remark 5.6

The asymptotic variance \(\gamma ^2=0\) when \(\sigma =\iota _m=1\cdots m\), in which case \(b=m\) and all blocks have length 1. This can be seen directly, since all other patterns occur only \(O_{\mathrm {p}}(n^{m-1})\) times (by Theorem 5.3), and thus \(\iota _m\) occurs \(\left( {\begin{array}{c}n\\ m\end{array}}\right) - O_{\mathrm {p}}(n^{m-1})\) times. This argument also shows that the asymptotic variance of \(n_{1\cdots m}(\varvec{\pi }_{{{231}, {312}};n})\) is of the order \(n^{2m-3}\).

It follows from Proposition 2.4 that \(\gamma ^2>0\) for any other \(\sigma \in \mathfrak {S}_*({231}, {312})\).

## 6 Avoiding \(\{{231}, {321}\}\)

In this section we consider \(T=\{{231}, {321}\}\). Equivalent sets are \(\{123, 132\}\), \(\{123, 213\}\), \(\{312, 321\}\).

It was shown by Simion and Schmidt [20] that \(|\mathfrak {S}_n({231}, {321})|=2^{n-1}\), together with the following characterization (in an equivalent form).

### Proposition 6.1

([20, Proposition 12]) A permutation \(\pi \) belongs to the class \(\mathfrak {S}_*({231}, {321})\) if and only if every block in \(\pi \) is of the type \(\ell 12\cdots (\ell -1)\) for some \(\ell \). \(\square \)

Thus, as in Sect. 5, a permutation in \(\mathfrak {S}_*({231}, {321})\) is determined by its sequence of block lengths, and these can be arbitrary. In this section, let \(\pi _{\ell _1,\dots ,\ell _b}\) denote the permutation in \(\mathfrak {S}_*({231}, {321})\) with block lengths \(\ell _1,\dots ,\ell _b\).

*R*counts the occurrences where less than

*b*different blocks in \(\pi _{L_1,\dots ,L_B}\) are used. We represent the block lengths as in Sect. 5, in particular (5.3)–(5.4), again using an infinite i.i.d. sequence \(X_i\sim \mathrm{Ge}(\frac{1}{2})\). Then, the main term in (6.1) is sandwiched between

*U*-statistics as in (5.6), and we can apply Proposition 2.4 to it. (Alternatively, we can use Proposition 2.5 as in Remark 5.2.)

### Theorem 6.2

Moreover, (6.9) holds with convergence of all moments.

### Proof

The argument above yields the stated limit for the first (main) term on the right-hand side of (6.1). We show that the remainder term *R* is negligible.

The term *R* can be split up as a sum \(\sum _{d=1}^{b-1} R_d\), where \(R_d\) counts the occurences that use *d* blocks in \(\pi =\pi _{L_1,\dots ,L_B}\). Each \(R_d\) may be written as a sum over *d*-tuples of blocks, and thus bounded as in (5.6) by some \(U\)-statistics \(U^{(d)}_{N_{+}(n)}\) of order *d*. Applying Proposition 2.4 (or Proposition 2.1, together with \(N_{+}(n)\leqslant n\)) to the latter, we find \(R_d=O_{\mathrm {p}}(n^d)=O_{\mathrm {p}}(n^{b-1})\), and thus \(R_d/n^{b-1/2}\overset{\mathrm {p}}{\longrightarrow }0\). For moments, we similarly have by Proposition 2.6 or 2.1\({\mathbb {E}{}}|R_d|^p = O\bigl (n^{pd}\bigr )=O\bigl (n^{p(b-1)}\bigr )=o\bigl (n^{p(b-1/2)}\bigr )\). Hence, each \(R_d\) is negligible in the limit (6.9), and the result follows. \(\square \)

### Example 6.3

*ij*with \(i<j\) equals \(\xi _j\), so the total number is \(\sum _2^n\xi _i\sim \mathrm{Bi}(n-1,\frac{1}{2})\).

### Remark 6.4

Unlike in Sect. 5, here the asymptotic mean depends not only on the number of blocks in \(\sigma \), but also on their lengths.

### Remark 6.5

As in Sect. 5, the asymptotic variance \(\gamma ^2=0\) when \(\sigma =\iota _m=1\cdots m\), in which case \(b=m\) and all blocks have length 1, but \(\gamma ^2>0\) for any other \(\sigma \in \mathfrak {S}_*({231}, {321})\).

## 7 Avoiding \(\{{132}, {321}\}\)

In this section we consider \(T=\{{132}, {321}\}\). Equivalent sets are \(\{123, 231\}\), \(\{123, 312\}\), \(\{213, 321\}\).

*k*, \(\ell \),

*m*(where the third run is empty when \(m=0\)).

### Proposition 7.1

*K*and

*L*are random with (

*K*,

*L*) uniformly distributed over the set \(\{K,L\geqslant 1:K+L\leqslant n\}\). As \({n\rightarrow \infty }\), we thus have \((K/n,L/n)\overset{\mathrm {d}}{\longrightarrow }(X,Y)\) with (

*X*,

*Y*) uniformly distributed on the triangle \(\{(X,Y)\in \mathbb {R}_+^2:X+Y\leqslant 1\}\). Equivalently, letting \(Z:=1-X-Y\),

*i*,

*j*,

*p*, then it is easily seen that an occurrence of \(\sigma \) in \(\pi _{k,\ell ,m}\) is obtained by selecting

*i*,

*j*and

*p*elements from the three runs of \(\pi _{k,\ell ,m}\), and thus

*i*elements from either the union of the first and last run, or from the union of the two last. Hence, by inclusion-exclusion,

### Theorem 7.2

- (i)If \(\sigma =\pi _{i,j,p}\) for some
*i*,*j*,*p*, thenwhere \((X,Y,Z)\sim \mathrm{Dir}(1,1,1)\).$$\begin{aligned} n^{-(i+j+p)}n_\sigma (\varvec{\pi }_{{{132}, {321}};n}) \overset{\mathrm {d}}{\longrightarrow }W_{i,j,p}:=\frac{1}{i!\,j!\,p!} X^i Y^jZ^p, \end{aligned}$$(7.6) - (ii)If \(\sigma =\iota _{i}\), thenwith \((X,Y,Z)\sim \mathrm{Dir}(1,1,1)\) as in (i).$$\begin{aligned} n^{-i}n_\sigma (\varvec{\pi }_{{{132}, {321}};n}) \overset{\mathrm {d}}{\longrightarrow }W_i:=\frac{1}{i!}\bigl ((X+Z)^i+(Y+Z)^i-Z^i\bigr ), \end{aligned}$$(7.7)

### Proof

Higher moments of \(W_{i,j,p}\) follow also from (7.10).

### Corollary 7.3

*X*,

*Y*) as above; the limit variable

*W*has density function

### Proof

We have \(21=\pi _{1,1,0}\), and thus (7.6) yields (7.11). The formula (7.13) for the moments \({\mathbb {E}{}}W^r={\mathbb {E}{}}X^rY^r\) follow by (7.10). Finally, for \(0<t<1/4\), \({\mathbb {P}{}}(W>t)={\mathbb {P}{}}(XY>t)\) equals 2 times the area of the set \(\{(x,y)\in \mathbb {R}_+^2:x+y\leqslant 1,\,xy> t\}\). A differentiation and a simple calculation yield (7.12). \(\square \)

### Example 7.4

*n*.) Note that by (7.8), all \(W_{i,j,q}\) with the same \(i+j+q\) have the same expectation; their distributions differ, however, in general, as is shown by higher moments. For example, in the present example, by (7.10), \({\mathbb {E}{}}W_{1,1,1}^2=2/7!\) and \({\mathbb {E}{}}W_{2,1,0}^2=3/7!\).

The expected number of occurrences of \(\sigma \) can also easily be found exactly for finite *n*, as follows. As noted above, (7.8) shows that all \(\sigma \) in (i) of the same length occur in \(\varvec{\pi }_{{{132}, {321}};n}\) with asymptotically equal frequencies. In fact, this holds also exactly, for any *n*. (Note also that (7.8) is an immediate consequence of (7.18).)

### Theorem 7.5

*n*,

### Proof

*k*and all \(q'_r\) by 1, and \(\ell \) and all \(q''_s\) by 2, we obtain a bijection with the collection of all subsets of \(i+j+q+2\) elements of \(\{1,\dots ,n+2\}\). Hence, the total number of occurrences is \(\left( {\begin{array}{c}n+2\\ i+j+p+2\end{array}}\right) \), and (7.18) follows. \(\square \)

## 8 Avoiding \(\{{231},{312}, {321}\}\)

We proceed to avoiding sets of three permutations. In this section we avoid \(T=\{{231},{312}, {321}\}\). An equivalent set is \(\{123,132,213\}\).

It was shown by Simion and Schmidt [20] that \(|\mathfrak {S}_n({231},{312}, {321})|=F_{n+1}\), the \((n+1)\hbox {th}\) Fibonacci number (with the initial conditions \(F_0=0\), \(F_1=1\)); they also gave the following characterization (in an equivalent form).

### Proposition 8.1

[20, Proposition \(15^*\)] A permutation \(\pi \) belongs to the class \(\mathfrak {S}_*({231},{312}, {321})\) if and only if every block in \(\pi \) is decreasing and has length \(\leqslant 2\), i.e., every block is 1 or 21. \(\square \)

Cf. Proposition 5.1; we have here added the restriction that block lengths are 1 or 2. With this restriction in mind, we use again the notation (5.1) and note that (5.2) holds. A permutation \(\pi \in \mathfrak {S}_n({231},{312}, {321})\) is thus of the form \(\pi _{L_1,\dots ,L_B}\) for some sequence \(L_1,\dots ,L_B\) of \(\{1,2\}\) with sum *n*; furthermore, this yields a bijection with all such sequences.

*p*to be the (inverse) golden ratio:

*X*be a random variable with the distribution

*X*, and let \(S_n:=\sum _{i=1}^nX_i\). Then for any sequence \(\ell _1,\dots ,\ell _b\) with \(b\geqslant 1\), \(\ell _i\in \{1,2\}\) and \(\sum _{1}^b\ell _i=n\),

*n*; we have seen above that this equals the distribution of the sequence of block lengths \((L_1,\dots ,L_B)\) of a random permutation \(\varvec{\pi }_{{{231},{312}, {321}};n}\) in \(\mathfrak {S}_n({231},{312}, {321})\). Consequently, recalling (2.14) and Remark 2.3,

*f*is defined by (5.5), then \(n_\sigma (\varvec{\pi }_{{{231},{312}, {321}};n})\) has the same distribution as \(U_{N_{+}(n)}\) conditioned on \(S_{N_{+}(n)}=n\). Consequently, Proposition 2.5 applies and yields asymptotic normality of \(n_\sigma (\varvec{\pi }_{{{231},{312}, {321}};n})\), and Proposition 2.6 adds moment convergence.

### Theorem 8.2

Moreover, (8.12) holds with convergence of all moments. \(\square \)

### Example 8.3

### Remark 8.4

Again, \(\gamma ^2>0\) unless \(\sigma =\iota _m\).

## 9 Avoiding \(\{{132},{231}, {312}\}\)

In this section we avoid \(\{{132},{231}, {312}\}\). Equivalent sets are \(\{132,213,231\}\), \(\{132,213,312\}\), \(\{213,231,312\}\).

### Proposition 9.1

Cf. Propositions 4.1 and 5.1, which characterize supersets. (Equivalently, \(\pi \in \mathfrak {S}_*({132},{231}, {312})\) if the first block is decreasing and all other blocks have length 1.)

### Theorem 9.2

- (i)If \(\sigma =\pi _{k,m-k}\) with \(2\leqslant k\leqslant m\), then$$\begin{aligned} n^{-m}n_\sigma (\varvec{\pi }_{{{132},{231}, {312}};n}) \overset{\mathrm {d}}{\longrightarrow }W_{k,m-k}:=\frac{1}{k!\,(m-k)!} U^k (1-U)^{m-k}. \end{aligned}$$(9.4)
- (ii)If \(\sigma =\pi _{1,m-1}=\iota _{m}\), then$$\begin{aligned} \begin{aligned} n^{-m}n_\sigma (\varvec{\pi }_{{{132},{231}, {312}};n}) \overset{\mathrm {d}}{\longrightarrow }W_{1,m-1}&:=\frac{1}{(m-1)!}U(1-U)^{m-1}+\frac{1}{m!}(1-U)^m\\&\phantom {:} =\frac{1}{m!}\bigl (1+(m-1)U\bigr )(1-U)^{m-1}. \end{aligned} \end{aligned}$$(9.5)

### Proof

The limits in distribution (9.4) and (9.5) hold (with joint convergence) by (9.3) and (9.2). Moment convergence holds because the normalized variables in (9.4) and (9.5) are bounded (by 1). Finally, the expectations in (9.6)–(9.7) are computed by standard beta integrals. \(\square \)

### Corollary 9.3

*W*has moments

## 10 Avoiding \(\{{132},{231}, {321}\}\)

In this section we avoid \(\{{132},{231}, {321}\}\). Equivalent sets are \(\{123,132,231\}\), \(\{123,213,312\}\), \(\{213,312,321\}\), \(\{123,132,312\}\), \(\{123,213,231\}\), \(\{132,312,321\}\), \(\{213,231,321\}\).

### Proposition 10.1

Cf. Proposition 7.1, which characterizes a superset.

### Theorem 10.2

- (i)If \(\sigma =\pi _{k,m-k}\) with \(2\leqslant k\leqslant m\), then$$\begin{aligned} n^{-(m-1)}n_\sigma (\varvec{\pi }_{{{132},{231}, {321}};n}) \overset{\mathrm {d}}{\longrightarrow }W_{k,m-k}:=\frac{1}{(k-1)!\,(m-k)!} U^{k-1} (1-U)^{m-k}.\nonumber \\ \end{aligned}$$(10.3)
- (ii)If \(\sigma =\pi _{1,m-1}=\iota _{m}\), then$$\begin{aligned} \begin{aligned} n^{-m}n_\sigma (\varvec{\pi }_{{{132},{231}, {321}};n}) =\frac{1}{m!}+O\bigl (n^{-1}\bigr ) \overset{\mathrm {p}}{\longrightarrow }\frac{1}{m!}. \end{aligned} \end{aligned}$$(10.4)

### Corollary 10.3

## 11 Avoiding \(\{{132},{213}, {321}\}\)

In this section we avoid \(\{{132},{213}, {321}\}\). An equivalent sets is \(\{123,231,312\}\).

### Proposition 11.1

Cf. Proposition 7.1, which again characterizes a superset.

### Theorem 11.2

- (i)If \(\sigma =\pi _{k,m-k}\) with \(1\leqslant k\leqslant m-1\), then$$\begin{aligned} n^{-m}n_\sigma (\varvec{\pi }_{{{132},{213}, {321}};n}) \overset{\mathrm {d}}{\longrightarrow }W_{k,m-k}:=\frac{1}{k!\,(m-k)!} U^{k} (1-U)^{m-k}. \end{aligned}$$(11.3)
- (ii)If \(\sigma =\pi _{m,0}=\iota _{m}\), then$$\begin{aligned} n^{-m}n_\sigma (\varvec{\pi }_{{{132},{213}, {321}};n}) \overset{\mathrm {d}}{\longrightarrow }W_{m,0}:=\frac{1}{m!} \bigl (U^{m}+ (1-U)^{m}\bigr ). \end{aligned}$$(11.4)

### Corollary 11.3

*W*has moments

## Notes

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