Patterns in random permutations avoiding some sets of multiple patterns

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.

The classes S * (τ ) and, more generally, S * (T ) have been studied for a long time, see e.g. Knuth [16,, Simion and Schmidt [20], Bóna [3]. In particular, one classical problem is to enumerate the sets S n (τ ), Date: 17 April, 2018. 2010 Mathematics Subject Classification. 60C05; 05A05, 05A16, 60F05. Partly supported by the Knut and Alice Wallenberg Foundation. 1 either exactly or asymptotically, see Bóna [3,. We note the fact that for any τ with length |τ | = 3, S n (τ ) has the same size |S n (τ )| = C n := 1 n+1 2n n , the n-th Catalan number, see e.g. [16,5], [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 |τ | 4 is more complicated.) The general problem that concerns us is to take a fixed set T of one or several permutations and let π T ;n be a uniformly random T -avoiding permutation, i.e., a uniformly random element of S n (T ), and then study the distribution of the random variable n σ (π T ;n ) for some other fixed permutation σ. (Only σ that are themselves T -avoiding are interesting, since otherwise n σ (π T ;n ) = 0.) One instance of this problem was studied already by Robertson, Wilf and Zeilberger [19], who gave a generating function for n 123 (π 132;n ). The exact distribution of n σ (π τ ;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 σ (π T ;n ), and of its moments, as n → ∞, for some fixed T and σ.
In the present paper we study the cases when T is a set of two or more permutations of length 3. The cases when T = {τ } for a single permutation τ of length |τ | = 3 were studied in [12; 13] (by symmetries, see Section 2.2, only two such cases have to be considered), and the cases when T contains a permutation of length 2 are trivial (there is then at most one permutation in S n (T ) for any n); hence the present paper completes the study of forbidding one or several permutations of length 3. The case of forbidding one or several permutations of length 4 seems much more complicated, but there are recent impressive results in some cases by Bassino, Bouvel, Féray, Gerin and Pierrot [2] and Bassino, Bouvel, Féray, Gerin, Maazoun and Pierrot [1].
The expectation E n σ (π T ;n ), or equivalently, the total number of occurences of σ in all T -avoiding permutations, has previously been treated in a number of papers for various cases, beginning with Bóna [5;7] (with τ = 132). In particular, Zhao [22] has given exact formulas when |σ| = 3 for the (non-trivial) cases treated in the present paper, where T consist of two or more permutations of length 3. Remark 1.1. For the non-restricted case of uniformly random permutations in S n , it is well-known that if π n is a uniformly random permutation in S n , then n σ (π n ) has an asymptotic normal distribution as n → ∞ for every fixed permutation σ; more precisely, if |σ| = m then, as n → ∞, for some γ 2 > 0 depending on σ; see Bóna [4;6] and Janson, Nakamura and Zeilberger [15,Theorem 4.1]. We obtain below similar asymptotic normal results in several cases (Sections 4,5,6,8); note that the asymptotic normality in particular implies concentration in these cases, in the sense On the other hand, in other cases (Sections 3, 7, 9, 10, 11) we find a different type of limits, where n σ (π T ;n )/ E n σ (π T ;n ) converges to some non-trivial positive random variable. The same holds in the case T = {2413, 3142} studied by Bassino, Bouvel, Féray, Gerin and Pierrot [2]. We see no obvious pattern in the occurence of these two types of limits in the cases below; nor do we know whether these are the only possibilities for a general set T of forbidden permutations. Remark 1.2. In the present paper we consider for simplicity often only univariate limits; corresponding multivariate results for several σ 1 , . . . , σ 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 1.3. In the present paper we study only the numbers n σ of occurences of some pattern in π τ ;n . There is also a number of papers by various authors that study other properties of random τ -avoiding permutations, see e.g. the references in [13]; such results will not be considered here.
Let π = π 1 · · · π n be a permutation. We say that a value π i is a maximum if π i > π j for every j < i, and a minimum if π i < π j for every j < i. (These are sometimes called LR maximum and LR minimum.) Note that π 1 always is both a maximum and a minimum.
For a set T of permutations we define T s := {τ s : τ ∈ T }. It follows from (2.4) that S n (T s ) = {π s : π ∈ S n (T )}, (2.5) and, furthermore, that for any permutation σ, (2.6) We say that the sets of forbidden permutations T and T s are equivalent, and note that (2.6) implies that it suffices to consider one set T in each equivalence class {T s : s ∈ 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 ι n avoids T .)

Compositions and decompositions of permutations.
If σ ∈ S m and τ ∈ S n , their composition σ * τ ∈ S m+n is defined by letting τ act on [m + 1, m + n] in the natural way; more formally, σ * τ = π ∈ S m+n where π i = σ i for 1 i m, and π j+m = τ j + m for 1 j n. It is easily seen that * is an associative operation that makes S * into a semigroup (without unit, since we only consider permutations of length 1). We say that a permutation π ∈ S * is decomposable if π = σ * τ for some σ, τ ∈ S * , and indecomposable otherwise; we also call an indecomposable permutation a block. Equivalently, π ∈ S n is decomposable if and only if π : [m] → [m] for some 1 m < n. See e.g. [8,Exercise VI.14].
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. An (asymmetric) U -statistic is a random variable of the form where X 1 , X 2 , . . . is an i.i.d. sequence of random variables and f is a given function of d 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. We shall use the central limit theorem for U -statistics, originally due to Hoeffding [9], in the asymmetric version given in [10,Theorem 11.20] and [14, Corollary 3.5 and (moment convergence) Theorem 3.15]. Let, with X denoting a generic X i , 10) [14] is f i (x) − µ in the present notation. 10; 14]). Suppose that f (X 1 , . . . , X d ) ∈ L 2 . Then, with the notation in (2.8)-(2.11), as n → ∞, (2.12) Moreover, if f (X 1 , . . . , X d ) ∈ L p for some p 2, the (2.12) holds with convergence of all moments of order p.
Example 2.2. A uniformly random permutation π n of length n (without other restrictions) can be constructed as the relative order of X 1 , . . . , X n , where X i are i.i.d. with, for example, a uniform distribution U(0, 1). For any given permutation σ ∈ S m , we can then write n σ (π n ) as a U -statistic (2.7) for a suitable indicator function f . Then Proposition 2.1 yields a limit theorem showing that n σ (π n ) is asymptotically normal. See [15] for details.
We shall also use a renewal theory version of Proposition 2.1. With the notations above, assume (for simplicity) that X i 0. Define S n := n i=1 X i , and let for each x > 0 N − (x) := sup{n : S n < x}, (2.13) N + (x) := inf{n : S n x} = N − (x) + 1. (2.14) Remark 2.3. The definitions (2.13)-(2.14) differ slightly from the ones in [14], where instead S n x and S n > x are used. This does not affect the asymptotic results used here. Note that the event {S k = n for some k 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 ± (x) means either N − (x) or N + (x); the results holds for both.

Proposition 2.5 ([14]). Suppose in addition to the hypotheses in Proposition 2.4 that X is integer-valued. Then (2.15) holds also conditioned on
Proposition 2.6 ( [14]). Suppose in addition to the hypotheses in Proposition 2.4 or 2.5 that f (X 1 , . . . , X d ) ∈ L p and X ∈ L p for every p < ∞. Then the conclusion (2.15) holds with convergence of all moments.

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 Subsection 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 e(x) is a random non-negative function on [0, 1].
The limit variables Λ σ in Theorem 3.1 can be expressed as functionals of a Brownian excursion e(x), see [12]; the description is, in general, rather complicated, but some cases are simple.
For the number n 21 of inversions, we thus have By Subsection 2.2, the left-hand side can also be seen as the number of inversions n 21 (π 231;n ) or n 21 (π 312;n ), normalized by n 3/2 , where we instead avoid 231 or 312.
for a positive random variable W σ that can be represented as where w σ is positive constant. Moreover, the convergence (3.5) holds jointly for any set of σ ∈ S * (321), and with convergence of all moments.
Again, in an occurrence of σ in π, each block in σ has to be mapped into a block in π. However, this time, several consecutive blocks in σ may be mapped to the same block in π, provided they have length 1. Moreover, if a block of length ℓ 2 in σ is mapped to a block in π, then the first element has to be mapped to the first element. Hence, we obtain instead of (5.2), if σ = π ℓ 1 ,...,ℓ b , and R counts the occurrences where less that b different blocks in π L 1 ,...,L B are used. We represent the block lengths as in Section 5, in particular (5.3)-(5.4), again using an infinite i.i.d. sequence X i ∼ Ge( 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.) By (5.7), E z X−1 = (2 − z) −1 , and calculations similar to (5.8) yield Simple calculations then yield, in addition to (5.13)-(5.14), letting b 1 be the number of blocks of length 1, Consequently, we obtain by Propositions 2.4 and 2.6 asymptotic normality in the following form.
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.
As n → ∞, we thus have (K/n, L/n) where we recall that the Dirichlet distribution Dir(1, 1, 1) is the uniform distribution on the simplex {(x, y, z) ∈ R 3 + : x + y + z = 1}. If σ = π i,j,p for some i, j, p, then it is easily seen that an occurrence of σ in π k,ℓ,m is obtained by selecting i, j and p elements from the three runs of π k,ℓ,m , and thus n σ (π k,ℓ,m ) = k i ℓ j m p . (7.4) Similarly, if σ = ι i , then an occurrence of σ in π k,ℓ,m is obtained by selecting i elements from either the union of the first and last run, or from the union of the two last. Hence, by inclusion-exclusion, These exact formulas together with the description of π 132,321;n above and (7.3) yield the following asymptotic result.
with (X, Y, Z) ∼ Dir(1, 1, 1) as in (i). Moreover, these hold jointly for any set of such σ, and with convergence of all moments. In particular, in case (i), and in case (ii), Proof. The limits in distribution (7.6) and (7.7) hold (with joint convergence) by the discussion before the theorem. Moment convergence holds because the normalized variables in (7.6) and (7.7) are bounded (by 1). Finally, the expectation in (7.8) is easily computed using the multidimensional extension of the beta integral [17, (5.14. 2)], which implies For the expectation in (7.9), we note also that X + Z d = Y + Z ∼ B(2, 1); the result follows by a short calculation.
Higher moments of W i,j,p follow also from (7.10). with (X, Y ) as above; the limit variable W has density function and moments Proof. We have 21 = π 1,1,0 , and thus (7.6) yields (7.11). The formula (11.8) for the moments E W r = E X r Y r follow by (7.10). Finally, for 0 < t < 1/4, P(W > t) = P(XY > t) equals 2 times the area of the set {(x, y) ∈ R 2 + : x + y 1, xy > t}. A differentiation and a simple calculation yield (7.12).