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Annals of Combinatorics

, Volume 23, Issue 3–4, pp 785–800 | Cite as

On Pattern-Avoiding Fishburn Permutations

  • Juan B. GilEmail author
  • Michael D. Weiner
Article
  • 31 Downloads

Abstract

The class of permutations that avoid the bivincular pattern \((231, \{1\},\{1\})\) is known to be enumerated by the Fishburn numbers. In this paper, we call them Fishburn permutations and study their pattern avoidance. For classical patterns of size 3, we give a complete enumerative picture for regular and indecomposable Fishburn permutations. For patterns of size 4, we focus on a Wilf equivalence class of Fishburn permutations that are enumerated by the Catalan numbers. In addition, we also discuss a class enumerated by the binomial transform of the Catalan numbers and give conjectures for other equivalence classes of pattern-avoiding Fishburn permutations.

Keywords

Pattern avoiding permutation Fishburn number Bivincular pattern 

Mathematics Subject Classification

Primary 05A05 Secondary 05A15 05A19 

1 Introduction

Motivated by a recent paper by Andrews and Sellers [1], we became interested in the Fishburn numbers \(\xi (n)\), defined by the formal power series
$$\begin{aligned} \sum _{n=0}^\infty \xi (n) q^n = 1+\sum _{n=1}^{\infty }\prod _{j=1}^n (1-(1-q)^j). \end{aligned}$$
They are listed as Sequence A022493 in [7] and have several combinatorial interpretations. For example, \(\xi (n)\) gives the:
\(\triangleright \)

number of linearized chord diagrams of degree n,

\(\triangleright \)

number of unlabeled \((2+2)\)-free posets on n elements,

\(\triangleright \)

number of ascent sequences of length n,

\(\triangleright \)

number of permutations in \(S_n\) that avoid a certain bivincular pattern.1

For more information on these interpretations, we refer the reader to [2] and the references therein.
In this note, we are primarily concerned with the aforementioned class of permutations. That they are enumerated by the Fishburn numbers was proved in [2] by Bousquet-Mélou, Claesson, Dukes, and Kitaev, where the authors introduced bivincular patterns (permutations with restrictions on the adjacency of positions and values) and gave a bijection to ascent sequences. More specifically, a permutation \(\pi \in S_n\) is said to contain the bivincular pattern \((231,\{1\},\{1\})\) if there are positions \(i<k\) with \(\pi (i)>1\), \(\pi (k) = \pi (i)-1\), such that the subsequence \(\pi (i)\pi (i+1)\pi (k)\) forms a 231 pattern. Such a bivincular pattern may be visualized by the plot
where bold lines indicate adjacent entries and gray lines indicate an elastic distance between the entries.
We let \({\mathscr {F}}_n\) denote the class of permutations in \(S_n\) that avoid the pattern Open image in new window, and since \(|{\mathscr {F}}_n|=\xi (n)\) (see [2]), we call the elements of \({\mathscr {F}}= \bigcup _n {\mathscr {F}}_n\)Fishburn permutations. Further, we let \({\mathscr {F}}_n(\sigma )\) denote the class of Fishburn permutations in \({\mathscr {F}}_n\) that avoid the pattern \(\sigma \).
Table 1

\(\sigma \)-avoiding Fishburn permutations

Pattern \(\sigma \)

\(|{\mathscr {F}}_n(\sigma )|\)

OEIS

123, 132, 213, 312

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...

A000079

231

1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...

A000108

321

1, 2, 4, 9, 22, 57, 154, 429, 1223, 3550, ...

A105633

Our goal is to study \(F_n(\sigma )=\left| {\mathscr {F}}_n(\sigma )\right| \) for classical patterns of size 3 or 4. In Sect. 2, we give a complete picture for regular and indecomposable Fishburn permutations that avoid a classical pattern of size 3. Table 1 and Table 2 provide a summary of our findings. In Sect. 3, we discuss patterns of size 4, focusing on a Wilf equivalence class of Fishburn permutations that are enumerated by the Catalan numbers \(C_n\) (see Table 4). We also prove the formula \( F_n(1342) = \sum _{k=1}^n \left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) C_{n-k}\), and conjecture two other equivalence classes (see Table 3). Finally, in Sect. 4, we briefly discuss indecomposable Fishburn permutations that avoid a pattern of size 4. In Table 5, we make some conjectures based on preliminary computations.
Table 2

\(\sigma \)-avoiding indecomposable Fishburn permutations

Pattern \(\sigma \)

\(|{\mathscr {F}}_n^{\textsf {ind}}(\sigma )|\)

OEIS

123

1, 1, 2, 5, 12, 27, 58, 121, 248, 503, ...

A000325

132, 213

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...

A000079

231

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...

A000108

312

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

A000012

321

1, 1, 1, 2, 5, 13, 35, 97, 275, 794, ...

A082582

Table 3

Equivalence classes with a single pattern

Pattern \(\sigma \)

\(|{\mathscr {F}}_n(\sigma )|\)

OEIS

1342

1, 2, 5, 15, 51, 188, 731, 2950, ...

A007317

1432

1, 2, 5, 14, 43, 142, 495, 1796, ...

 

2314

1, 2, 5, 15, 52, 200, 827, 3601, ...

 

2341

1, 2, 5, 15, 52, 202, 858, 3910, ...

 

3412

1, 2, 5, 15, 52, 201, 843, 3764, ...

A202062(?)

3421

1, 2, 5, 15, 52, 203, 874, 4076, ...

 

4123

1, 2, 5, 14, 42, 133, 442, 1535, ...

 

4231

1, 2, 5, 15, 52, 201, 843, 3765, ...

 

4312

1, 2, 5, 14, 43, 143, 508, 1905, ...

 

4321

1, 2, 5, 14, 45, 162, 639, 2713, ...

 
Table 4

Catalan equivalent class

Pattern \(\sigma \)

\(|{\mathscr {F}}_n(\sigma )|\)

OEIS

1234, 1243, 1324, 1423, 2134, 2143, 3124, 3142

1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...

A000108

Basic notation. Permutations will be written in one-line notation. Given two permutations \(\sigma \) and \(\tau \) of sizes k and \(\ell \), respectively, their direct sum \(\sigma \oplus \tau \) is the permutation of size \(k + \ell \) consisting of \(\sigma \) followed by a shifted copy of \(\tau \). Similarly, their skew sum \(\sigma \ominus \tau \) is the permutation consisting of \(\tau \) preceded by a shifted copy of \(\sigma \). For example, \(312\oplus 21 = 31254\) and \(312\ominus 21 = 53421\).

A permutation is said to be indecomposable if it cannot be written as a direct sum of two nonempty permutations.

\(\mathsf {Av}_n(\sigma )\) denotes the class of permutations in \(S_n\) that avoid the pattern \(\sigma \). It is well known that if \(\sigma \in S_3\) then \(|\mathsf {Av}_n(\sigma )|=C_n\), where \(C_n\) is the Catalan number \(\frac{1}{n+1}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \), see e.g. [4].

2 Avoiding Patterns of Size 3

Clearly, \(\mathsf {Av}_n(231) \subset {\mathscr {F}}_n\). Now, since every Fishburn permutation that avoids the classical pattern 231 is contained in the set of 231-avoiding permutations, we get
$$\begin{aligned} {\mathscr {F}}_n(231) = \mathsf {Av}_n(231), \text { and so } F_n(231)=C_n. \end{aligned}$$
(2.1)
Enumeration of the Fishburn permutations that avoid the other five classical patterns of size 3 is less obvious.

Theorem 2.1

For \(\sigma \in \{123,132,213,312\}\), we have \(F_n(\sigma ) = 2^{n-1}\).

Proof

First of all, note that for every \(\sigma \) of size 3, we have \(F_1(\sigma )=1\) and \(F_2(\sigma )=2\).
Table 5

Some \(\sigma \)-avoiding indecomposable classes

Pattern \(\sigma \)

\(|{\mathscr {F}}_n^{\textsf {ind}}(\sigma )|\)

OEIS

1234

1, 1, 2, 6, 22, 85, 324, 1204, ...

 

1243, 2134

1, 1, 2, 6, 21, 75, 266, 938, ...

A289597(?)

1324

1, 1, 2, 6, 22, 84, 317, 1174, ...

 

1342

1, 1, 2, 6, 22, 88, 367, 1568, ...

A165538

1423, 3124

1, 1, 2, 6, 20, 68, 233, 805, ...

A279557

1432

1, 1, 2, 6, 20, 71, 263, 1002, ...

 

2143

1, 1, 2, 6, 19, 62, 207, 704, ...

A026012

2314

1, 1, 2, 6, 23, 99, 450, 2109, ...

 

2341

1, 1, 2, 6, 22, 91, 409, 1955, ...

 

2413, 2431, 3241

1, 1, 2, 6, 22, 90, 395, 1823, ...

A165546(?)

3142

1, 1, 2, 5, 14, 42, 132, 429, ...

A000108

3214

1, 1, 2, 6, 20, 72, 275, 1096, ...

 

3412

1, 1, 2, 6, 22, 90, 396, 1840, ...

 

3421

1, 1, 2, 6, 22, 92, 423, 2088, ...

 

4123

1, 1, 2, 5, 14, 43, 143, 507, ...

 

4132, 4213

1, 1, 2, 5, 15, 51, 188, 732, ...

 

4231

1, 1, 2, 6, 22, 90, 396, 1841, ...

 

4312

1, 1, 2, 5, 15, 51, 188, 733, ...

 

4321

1, 1, 2, 5, 17, 66, 279, 1256, ...

 
Case\(\sigma =132\): If \(\pi \in {\mathscr {F}}_{n-1}(132)\), the permutations \(1\ominus \pi \) and \(\pi \oplus 1\) are both in \({\mathscr {F}}_n(132)\). On the other hand, if \(\tau \) is a permutation in \(\mathsf {Av}_n(132)\) with \(\tau (i)=n\) for some \(1<i<n\), then we must have \(\tau (j)>\tau (k)\) for every \(j\in \{1,\dots ,i-1\}\) and \(k\in \{i+1,\dots ,n\}\). Thus, \(n-i=\tau (k')\) for some \(k'>i\) and \(n-i+1=\tau (j')\) for some \(j'<i\). But this violates the Fishburn condition since \(n-i+1\) is the smallest value to the left of n and must, therefore, be part of an ascent in \(\tau (1)\cdots \tau (i-1)\,n\). In other words, \({\mathscr {F}}_{n}(132)\) is the disjoint union of the sets \(\{1\ominus \pi : \pi \in {\mathscr {F}}_{n-1}(132)\}\) and \(\{\pi \oplus 1: \pi \in {\mathscr {F}}_{n-1}(132)\}\). Thus
$$\begin{aligned} F_n(132) = 2F_{n-1}(132)\end{aligned}$$
for \(n>1\),which implies \(F_n(132) = 2^{n-1}\).

Case\(\sigma =123\): For \(n>2\), the permutation \((n-1)(n-2)\cdots 21n\) is the only permutation in \({\mathscr {F}}_{n}(123)\) that ends with n, and if \(\pi \in {\mathscr {F}}_{n-1}(123)\), then \(1\ominus \pi \in {\mathscr {F}}_{n}(123)\).

Assume \(\tau \in {\mathscr {F}}_{n}(123)\) is such that \(\tau (i)=n\) for some \(1<i<n\). Since \(\tau \) avoids the pattern 123, we must have \(\tau (1)>\tau (2)>\cdots >\tau (i-1)\). Moreover, the Fishburn condition forces \(\tau (i-1)=1\), which implies \(\tau (i+1)>\tau (i+2)>\cdots >\tau (n)\). In other words, \(\tau \) may be any permutation with \(\tau (i-1)=1\), \(\tau (i)=n\) for which the entries to the left of 1 and to the right of n form two decreasing sequences. There are \(\left( {\begin{array}{c}n-2\\ i-2\end{array}}\right) \) such permutations.

In conclusion, we have the recurrence
$$\begin{aligned} F_n(123) = F_{n-1}(123) + 1 + \sum _{i=2}^{n-1}\left( {\begin{array}{c}n-2\\ i-2\end{array}}\right) = F_{n-1}(123) + 2^{n-2}, \end{aligned}$$
which implies \(F_n(123) = 2^{n-1}\).

Case\(\sigma =213\): For \(n>2\), the permutation \(12\cdots n\) is the only permutation in \({\mathscr {F}}_{n}(213)\) that ends with n, and if \(\pi \in {\mathscr {F}}_{n-1}(213)\), then \(1\ominus \pi \in {\mathscr {F}}_{n}(213)\).

Assume \(\tau \in {\mathscr {F}}_{n}(213)\) is such that \(\tau (i)=n\) for some \(1<i<n\). Since \(\tau \) avoids the pattern 213, we must have \(\tau (1)<\tau (2)<\cdots <\tau (i-1)\) and the Fishburn condition forces \(\tau (j)=j\) for every \(j\in \{1,\dots ,i-1\}\). Thus \(\tau \) must be of the form \(\tau = 1\cdots (i-1)n\pi _R\), where \(\pi _R\) may be any element of \({\mathscr {F}}_{n-i}(213)\). This implies
$$\begin{aligned} F_n(213) = 1 + \sum _{i=1}^{n-1} F_{n-i}(213), \end{aligned}$$
and we conclude \(F_n(213) = 2^{n-1}\).
Case\(\sigma =312\): If \(\pi \in {\mathscr {F}}_{n-1}(312)\), the permutation \(1\oplus \pi \) is in \({\mathscr {F}}_n(312)\). On the other hand, if \(\tau \) is a permutation in \(\mathsf {Av}_n(312)\) with \(\tau (i)=1\) for some \(1<i\le n\), then we must have \(\tau (j)<\tau (k)\) for every \(j\in \{1,\dots ,i-1\}\) and \(k\in \{i+1,\dots ,n\}\). Moreover, the Fishburn condition forces \(\tau (j)=i+1-j\) for every \(j\in \{1,\dots ,i-1\}\). Thus, \(\tau \) must be of the form \(\tau = i\cdots 21\pi _R\), where \(\pi _R=\emptyset \) if \(i=n\), or \(\pi _R\in {\mathscr {F}}_{n-i}(312)\) if \(i<n\). This implies
$$\begin{aligned} F_n(312) = 1 + \sum _{i=1}^{n-1} F_{n-i}(312), \end{aligned}$$
hence \(F_n(312) = 2^{n-1}\). \(\square \)

For our next result, we use a bijection between \(\mathsf {Av}_n(321)\) and the set of Dyck paths of semilength n, via the left-to-right maxima.2 Here, a Dyck path of semilength n is a simple lattice path from (0, 0) to (nn) that stays weakly above the diagonal \(y=x\) (with vertical unit steps U and horizontal unit steps D). On the other hand, a left-to-right maximum of a permutation \(\pi \) is an element \(\pi _i\) such that \(\pi _j<\pi _i\) for every \(j<i\).

The bijective map between \(\mathsf {Av}_n(321)\) and the set of Dyck paths of semilength n is defined as follows: Given \(\pi \in \mathsf {Av}_n(321)\), write
$$\begin{aligned} \pi = m_1w_1m_2w_2\cdots m_sw_s, \end{aligned}$$
where \(m_1,\dots ,m_s\) are the left-to-right maxima of \(\pi \), and each \(w_i\) is a subword of \(\pi \). Let \(|w_i|\) denote the length of \(w_i\). Reading the decomposition of \(\pi \) from left to right, we construct a path starting with \(m_1\)U-steps, \(|w_1|+1\)D-steps, and for every other subword \(m_iw_i\), we add \(m_i-m_{i-1}\)U-steps followed by \(|w_i|+1\)D-steps. In short, identify the left-to-right maxima in the plot of \(\pi \) and draw your path over them. For example, for \(\pi =351264 \in \mathsf {Av}_6(321)\) we get:
Note that \(\pi =351264 \not \in {\mathscr {F}}_6\).

Theorem 2.2

The set \({\mathscr {F}}_n(321)\) is in bijection with the set of Dyck paths of semilength n that avoid the subpath UUDU. By [6, Proposition 5] we then have
$$\begin{aligned} F_n(321) = \sum _{j=0}^{\lfloor (n-1)/2\rfloor } \frac{(-1)^j}{n-j}\left( {\begin{array}{c}n-j\\ j\end{array}}\right) \left( {\begin{array}{c}2n-3j\\ n-j+1\end{array}}\right) . \end{aligned}$$
This is Sequence [7, A105633].

Proof

Under the above bijection, an ascent \(\pi _i<\pi _{i+1}\) in \(\pi \in \mathsf {Av}_n(321)\) with \(k=\pi _{i+1}-\pi _i\) generates the subpath \(UDU^k\) in the corresponding Dyck path \(P_\pi \), and if \(\pi _i - 1=\pi _j\) for some \(j>i+1\), then \(P_\pi \) must necessarily contain the subpath \(UUDU^k\). Thus, we have that \(\pi \) avoids the pattern Open image in new window if and only if \(P_\pi \) avoids UUDU. \(\square \)

2.1 Indecomposable Permutations

Let \({\mathscr {F}}_n^{\textsf {ind}}(\sigma )\) be the set of indecomposable Fishburn permutations that avoid the pattern \(\sigma \), and let \(I\!F_n(\sigma )\) denote the number of elements in \({\mathscr {F}}_n^{\textsf {ind}}(\sigma )\). Observe that for every \(\sigma \) of size \(\ge 3\), we have \(I\!F_1(\sigma )=1\) and \(I\!F_2(\sigma )=1\).

We start with a fundamental known lemma, see e.g. [3, Lemma 3.1].

Lemma 2.3

If a pattern \(\sigma \) is indecomposable, then the sequence \(|\mathsf {Av}_n(\sigma )|\) is the invert transform of the sequence \(|\mathsf {Av}_n^{\textsf {ind}}(\sigma )|\). That is, if \(A^\sigma (x)\) and \(A_I^\sigma (x)\) are the corresponding generating functions, then
$$\begin{aligned} 1+A^\sigma (x) = \frac{1}{1-A_I^\sigma (x)}, \text { and so }\; A_I^\sigma (x)= \frac{A^\sigma (x)}{1+A^\sigma (x)}. \end{aligned}$$
In particular, since 231 is indecomposable, these identities are also valid for Fishburn permutations. The sequence \((I\!F_n)_{n\in {\mathbb {N}}}\) that enumerates indecomposable Fishburn permutations of size n starts with
$$\begin{aligned} 1, 1, 2, 6, 23, 104, 534, 3051, 19155, 130997, \dots . \end{aligned}$$

Theorem 2.4

For \(n>1\), we have \(I\!F_n(123) = 2^{n-1} - (n - 1)\).

Proof

As discussed in the proof of Theorem 2.1, for \(n>2\) the set \({\mathscr {F}}_{n}(123)\) consists of elements of the form \(1\ominus \pi \) with \(\pi \in {\mathscr {F}}_{n-1}(123)\), and elements of the form \(\tau = (i-1)\cdots 1n\pi _R\) for some \(1<i\le n\) and \(\pi _R\in {\mathscr {F}}_{n-i}(123)\). This forces \(\pi _R=\emptyset \) if \(i=n\), and \(\pi _R=(n-1)\cdots i\) if \(i<n\). Thus the only decomposable elements of \({\mathscr {F}}_n(123)\) are the \(n-1\) permutations
$$\begin{aligned} \begin{array}{c} 1n(n-1)\cdots 32, \\ 21n(n-1)\cdots 3, \\ \vdots \\ (n-1)\cdots 321n. \end{array} \end{aligned}$$
In conclusion, \(I\!F_n(123) = F_n(123)-(n - 1) = 2^{n-1} - (n - 1)\). \(\square \)

Theorem 2.5

For \(n>1\) and \(\sigma \in \{132,213\}\), we have \(I\!F_n(\sigma ) = 2^{n-2}\).

Proof

From the proof of Theorem 2.1, we know that for \(n>1\) every element of \({\mathscr {F}}_n(132)\) must be of the form \(1\ominus \pi \) or \(\pi \oplus 1\) with \(\pi \in {\mathscr {F}}_{n-1}(132)\). Since \(\pi \oplus 1\) is decomposable and \(1\ominus \pi \) is indecomposable, we have
$$\begin{aligned} {\mathscr {F}}_n^{\textsf {ind}}(132) = \{1\ominus \pi :\pi \in {\mathscr {F}}_{n-1}(132)\}. \end{aligned}$$
We also know that \({\mathscr {F}}_{n}(213)\) consists of elements of the form \(1\ominus \pi \) with \(\pi \in {\mathscr {F}}_{n-1}(213)\), and elements of the form \(\tau = (1\cdots (i-1)) \oplus (1\ominus \pi _R)\) for some \(1<i\le n\) (with \(\pi _R=\emptyset \) when \(i=n\)). Thus
$$\begin{aligned} {\mathscr {F}}_n^{\textsf {ind}}(213) = \{1\ominus \pi :\pi \in {\mathscr {F}}_{n-1}(213)\}. \end{aligned}$$
In conclusion, if \(\sigma \in \{132,213\}\), we have \(I\!F_n(\sigma ) = F_{n-1}(\sigma ) = 2^{n-2}\). \(\square \)

Let \(F^{\sigma }(x)\) and \( I\!F^{\sigma }(x)\) be the generating functions associated with the sequences \((F_n(\sigma ))_{n\in {\mathbb {N}}}\) and \((I\!F_n(\sigma ))_{n\in {\mathbb {N}}}\), respectively.

Theorem 2.6

For \(\sigma \in \{231,312,321\}\), we have
$$\begin{aligned} I\!F^{\sigma }(x) = \frac{F^{\sigma }(x)}{1+F^{\sigma }(x)}. \end{aligned}$$
In particular, \(I\!F_n(231) = C_{n-1} \) and \(I\!F_n(312) = 1\).

Proof

This follows from (2.1), Theorem 2.1, and Lemma 2.3. \(\square \)

Theorem 2.7

The sequence \(a_n = I\!F_n(321)\) satisfies the recurrence relation
$$\begin{aligned} a_n = a_{n-1} + \sum _{j=2}^{n-2} a_ja_{n-j} \end{aligned}$$
for \(n\ge 4\), with \(a_1=a_2=a_3=1\). This is Sequence [7, A082582].

Proof

We use the same Dyck path approach as in the proof of Theorem 2.2. Under this bijection, indecomposable permutations correspond to Dyck paths that do not touch the line \(y=x\) except at the end points.

Let \({\mathcal {A}}_n\) be the set of Dyck paths corresponding to \({\mathscr {F}}_n^{\textsf {ind}}(321)\). We will prove that \(a_n=|{\mathcal {A}}_n|\) satisfies the claimed recurrence relation. Clearly, for \(n=1,2,3\), the only indecomposable Fishburn permutations are 1, 21, and 312, which correspond to the Dyck paths UD, \(U^2D^2\), and \(U^3D^3\), respectively. Thus, \(a_1=a_2=a_3=1\).

Note that indecomposable permutations may not start with 1 or end with n. Moreover, every element of \(\pi \in {\mathscr {F}}_n^{\textsf {ind}}(321)\) must be of the form \(m1\pi (3)\cdots \pi (n)\) with \(m\ge 3\). Therefore, the elements of \({\mathcal {A}}_n\) have no peaks at the points (0, 1), (0, 2), or \((n-1,n)\), and for \(n>3\) their first return to the line \(y=x+1\) happens at a lattice point \((x,x+1)\) with \(x\in [2,n-1]\).

Dyck paths in \({\mathcal {A}}_n\) having \((n-1,n)\) as their first return to \(y=x+1\), are in one-to-one correspondence with the elements of \({\mathcal {A}}_{n-1}\) (just remove the first U and the last D of the longer path). Now, for \(j\in \{2,\dots ,n-2\}\), the set of paths \(P\in {\mathcal {A}}_n\) having first return to \(y=x+1\) at the point \((j,j+1)\) corresponds uniquely to the set of all pairs \((P_j,P_{n-j})\) with \(P_j\in {\mathcal {A}}_{j}\) and \(P_j\in {\mathcal {A}}_{n-j}\). For example,
This implies that there are \(a_ja_{n-j}\) paths in \({\mathcal {A}}_n\) having the point \((j,j+1)\) as their first return to the line \(y=x+1\). Finally, summing over j gives the claimed identity. \(\square \)

Here is a summary of our enumeration results for patterns of size 3:

3 Avoiding Patterns of Size 4

In this section, we discuss the enumeration of Fishburn permutations that avoid a pattern of size 4. There are at least 13 Wilf equivalence classes that we break down into three categories: 10 classes with a single pattern, 2 classes with (conjecturally) three patterns each, and a larger class with eight patterns enumerated by the Catalan numbers.

We will provide a proof for the enumeration of the class \({\mathscr {F}}_n(1342)\), but our main focus in this paper will be on the enumeration of the equivalence class given in Table 4.

For the remaining patterns we have the following conjectures.

Conjecture 3.1

\({\mathscr {F}}_n(2413)\sim {\mathscr {F}}_n(2431)\sim {\mathscr {F}}_n(3241)\).

Conjecture 3.2

\({\mathscr {F}}_n(3214)\sim {\mathscr {F}}_n(4132)\sim {\mathscr {F}}_n(4213)\).

Our first result of this section involves the binomial transform of the Catalan numbers, namely the sequence [7, A007317].

Theorem 3.3

$$\begin{aligned} F_n(1342) = \sum _{k=1}^n \left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) C_{n-k}. \end{aligned}$$

Proof

Let \({\mathcal {A}}_{n,k}\) be the set of all permutations \(\pi \in S_n\) such that
\(\circ \)

\(\pi (k)=1\) and \(\pi (1)>\pi (2)>\cdots >\pi (k-1)\),

\(\circ \)

\(\pi (k+1)\cdots \pi (n) \in \mathsf {Av}_{n-k}(231)\),

and let \({\mathscr {A}}_n = \bigcup _{k=1}^n {\mathcal {A}}_{n,k}\). Clearly,
$$\begin{aligned} |{\mathscr {A}}_n| = \sum _{k=1}^n |{\mathcal {A}}_{n,k}| = \sum _{k=1}^n\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) C_{n-k}. \end{aligned}$$
We will prove the theorem by showing that \({\mathscr {A}}_n = {\mathscr {F}}_n(1342)\).

First of all, since \({\mathcal {A}}_{n,k} \subset \mathsf {Av}_n(1342)\) and \(\mathsf {Av}_{n-k}(231) = {\mathscr {F}}_{n-k}(231)\) for every n and k, we have \({\mathscr {A}}_n\subset {\mathscr {F}}_n(1342)\).

Going in the other direction, let \(\pi \in {\mathscr {F}}_n(1342)\) and let k be such that \(\pi (k)=1\). Thus, \(\pi \) is of the form \(\pi = \pi (1)\cdots \pi (k-1) \,1\, \pi (k+1)\cdots \pi (n)\), which implies \(\pi (k+1)\cdots \pi (n) \in \mathsf {Av}_{n-k}(231)\). Now, if there is a \(j\in \{1,\dots ,k-2\}\) such that
$$\begin{aligned} \pi (1)>\cdots > \pi (j)<\pi (j+1), \end{aligned}$$
then \(\pi (j)-1=\pi (\ell )\) for some \(\ell >j+1\), and the pattern \(\pi (j)\pi (j+1)\pi (\ell )\) would violate the Fishburn condition. In other words, the entries to the left of \(\pi (k)=1\) must form a decreasing sequence, which implies \(\pi \in {\mathcal {A}}_{n,k}\subset {\mathscr {A}}_n\). Thus \({\mathscr {F}}_n(1342)\subset {\mathscr {A}}_n\), and we conclude that \({\mathscr {A}}_n = {\mathscr {F}}_n(1342)\). \(\square \)

3.1 Catalan Equivalence Class

The remaining part of this section is devoted to prove that \(|{\mathscr {F}}_n(\sigma )|=C_n\) for every \(\sigma \in \{1234, 1243, 1324, 1423, 2134, 2143, 3124, 3142\}\).

Theorem 3.4

We have \({\mathscr {F}}_n(3142)={\mathscr {F}}_n(231)\), hence \(F_n(3142) = C_n\).

Proof

Since 3142 contains the pattern 231, we have \({\mathscr {F}}_n(231)\subseteq {\mathscr {F}}_n(3142)\).

To prove the reverse inclusion, suppose that there exists \(\pi \in {\mathscr {F}}_n(3142)\) such that \(\pi \) contains the pattern 231. Let \(i<j<k\) be the positions of the 231 pattern such that
\(\circ \)

\(\pi (k)<\pi (i)<\pi (j)\),

\(\circ \)

\(\pi (i)\) is the left-most entry of \(\pi \) involved in a 231 pattern,

\(\circ \)

\(\pi (j)\) is the first entry with \(j>i\) such that \(\pi (i)<\pi (j)\),

\(\circ \)

\(\pi (k)\) is the largest entry with \(k>j\) such that \(\pi (k)<\pi (i)\).

In other words, assume the plot of \(\pi \) is of the form
where no elements of \(\pi \) may occur in the shaded regions. It follows that, if \(\ell \) is the position of \(\pi (k)+1\), then \(i\le \ell <j\). But this is not possible since, \(\pi (\ell )<\pi (\ell +1)\) violates the Fishburn condition, and \(\pi (\ell )>\pi (\ell +1)\) implies \(\pi (k)>\pi (\ell +1)\) which forces the existence of a 3142 pattern. In conclusion, no permutation \(\pi \in {\mathscr {F}}_n(3142)\) is allowed to contain a 231 pattern. Therefore, \({\mathscr {F}}_n(3142)\subseteq {\mathscr {F}}_n(231)\) and we obtain the claimed equality. \(\square \)

Theorem 3.5

\({\mathscr {F}}_n(1234)\sim {\mathscr {F}}_n(1243)\) and \({\mathscr {F}}_n(2134)\sim {\mathscr {F}}_n(2143)\).

Proof

To prove both Wilf equivalence relations, we use a bijection
$$\begin{aligned} \phi :\mathsf {Av}_n(\tau \oplus 12)\rightarrow \mathsf {Av}_n(\tau \oplus 21) \end{aligned}$$
given by West in [8], which we proceed to describe.
For \(\pi \in S_n\) and \(\sigma \in S_k\), \(k<n\), let \(B_{\pi }(\sigma )\) be the set of maximal values of all instances of the pattern \(\sigma \) in the permutation \(\pi \). For example, \(B_\pi (\sigma )=\emptyset \) if \(\pi \) avoids \(\sigma \), and for \(\pi =531968274\) we have \(B_\pi (123)=\{4,7,8\}\) (Fig. 1).
Fig. 1

\(\pi =531968274\) and \(B_\pi (123)=\{4,7,8\}\)

For \(\pi \in \mathsf {Av}_n(\tau \oplus 12)\), let \(\ell \) be the number of elements in \(B_\pi (\tau \oplus 1)\). If \(\ell =0\), we define \(\phi (\pi ) = \pi \). If \(\ell >0\), we let \(i_1<\dots <i_\ell \) be the positions in \(\pi \) of the elements of \(B_\pi (\tau \oplus 1)\) and define
$$\begin{aligned} {\tilde{\pi }}(j) = \pi (j) \;\text { if } j\not \in \{i_1,\dots ,i_\ell \}. \end{aligned}$$
Note that \(\pi \in \mathsf {Av}_n(\tau \oplus 12)\) implies \(\pi (i_1)>\cdots >\pi (i_\ell )\).
Let \(b_1\) be the smallest element of \(B_\pi (\tau \oplus 1)\) such that \({\tilde{\pi }}(1)\cdots {\tilde{\pi }}(i_1-1)b_1\) contains the pattern \(\tau \oplus 1\). Define
$$\begin{aligned} {\tilde{\pi }}(i_1) = b_1. \end{aligned}$$
Iteratively, for \(k=2,\dots ,\ell \), we let \(b_k\) be the smallest element of \(B_\pi (\tau \oplus 1)\backslash \{b_1,\dots ,b_{k-1}\}\) such that \({\tilde{\pi }}(1)\cdots {\tilde{\pi }}(i_k-1)b_k\) contains the pattern \(\tau \oplus 1\). We then define
$$\begin{aligned} {\tilde{\pi }}(i_k) = b_k \;\text { for } k=2,\dots ,\ell , \end{aligned}$$
to complete the definition of \({\tilde{\pi }} = \phi (\pi )\).
For example, for \(\pi =\text {53196} \mathbf{8}\text {2} \mathbf{74}\) and \(\tau =12\), we have \(\ell =3\)\(\tilde{\pi }= \text{53196} \mathbf{7}\text{2} \mathbf{48}\) (Fig. 2).
Fig. 2

\({\tilde{\pi }} = \phi (531968274) = 531967248\)

It is easy to check that the map \(\phi \) induces a bijection
$$\begin{aligned} \phi :{\mathscr {F}}_n(\tau \oplus 12)\rightarrow {\mathscr {F}}_n(\tau \oplus 21). \end{aligned}$$
Indeed, if \(\pi (i_k)\in B_\pi (\tau \oplus 1)\) is such that \(\pi (i_k-1)<\pi (i_k)\), then \({\tilde{\pi }}(i_k-1)=\pi (i_k-1)\) and the pair \({\tilde{\pi }}(i_k-1), {\tilde{\pi }}(i_k)\) does not create a pattern Open image in new window.

On the other hand, if \(\pi (i_1)>\dots >\pi (i_k)\) is a maximal descent of elements from \(B_\pi (\tau \oplus 1)\), and if \(\pi (i_j)-1>0\) (for \(j\in \{1,\dots ,k\}\)) is not part of that descent, then \(\pi (i_j)-1\) must be to the left of \(\pi (i_1)\) and so any ascent in \({\tilde{\pi }}(i_1)\cdots {\tilde{\pi }}(i_k)\) cannot create the pattern Open image in new window.

Thus, if \(\pi \in \mathsf {Av}_n(\tau \oplus 12)\) is Fishburn, so is \({\tilde{\pi }} = \phi (\pi )\in \mathsf {Av}_n(\tau \oplus 21)\). \(\square \)

Theorem 3.6

\({\mathscr {F}}_n(1423)\sim {\mathscr {F}}_n(1243)\sim {\mathscr {F}}_n(1234)\sim {\mathscr {F}}_n(1324)\).

Proof

Let \(\alpha :{\mathscr {F}}_n(1423)\rightarrow {\mathscr {F}}_n(1243)\) be the map defined through the following algorithm.

Algorithm\(\alpha \): Let \(\pi \in {\mathscr {F}}_n(1423)\) and set \({\tilde{\pi }}=\pi \).
  1. Step 1:

    If \({\tilde{\pi }} \not \in \mathsf {Av}_n(1243)\), let \(i<j<k<\ell \) be the positions of the left-most 1243 pattern contained in \({\tilde{\pi }}\). Redefine \({\tilde{\pi }}\) by moving \({\tilde{\pi }}(k)\) to position j, shifting the entries at positions j through \(k-1\) one step to the right.

     
  2. Step 2:

    If \({\tilde{\pi }} \in \mathsf {Av}_n(1243)\), then return \(\alpha (\pi )={\tilde{\pi }}\); otherwise go to Step 1.

     
For example, for \(\pi = 2135476 \in {\mathscr {F}}(1423)\), the above algorithm yields
$$\begin{aligned} \mathbf{2}\text{1} \mathbf{354}\text{76}&\longrightarrow \text{21} \mathbf{5}\text{3476} \not \in \mathsf {Av}(1243)\\&\swarrow \\ \mathbf{2}\text{1} \mathbf{5} \text{34} \mathbf{76}&\longrightarrow \text{21} \mathbf{7}\text{5346} \in \mathsf {Av}(1243) \end{aligned}$$
and so \(\alpha (\pi ) = 2175346\in {\mathscr {F}}(1243)\).

Observe that the map \(\alpha \) changes every 1243 pattern into a 1423 pattern. To see that it preserves the Fishburn condition, let \(\pi \in {\mathscr {F}}_n(1423)\) be such that \(\pi (i)\), \(\pi (j)\), \(\pi (k)\), \(\pi (\ell )\) form a left-most 1243 pattern. Thus, at first, \(\pi \) must be of the form

where no elements of \(\pi \) may occur in the shaded regions. In particular, we must have
$$\begin{aligned} \pi (j-1)<\pi (j) \quad \text {and}\quad \pi (k-1)<\pi (k). \end{aligned}$$
(3.1)
Hence the step of moving \(\pi (k)\) to position j does not create a new ascent and, therefore, it cannot create a pattern Open image in new window. After one iteration, \({\tilde{\pi }}\) takes the form

and if the left-most 1243 pattern \({\tilde{\pi }}(i)\), \({\tilde{\pi }}(j)\), \({\tilde{\pi }}(k)\), \({\tilde{\pi }}(\ell )\) contained in \({\tilde{\pi }}\) has its second entry at a position different from \(j'\), then \({\tilde{\pi }}\) must satisfy (3.1) and no pattern Open image in new window will be created.

Otherwise, if \(j=j'\), then either \(k=\ell '\) or \(\ell =\ell '\). In the first case, we have \({\tilde{\pi }}(k-1)<{\tilde{\pi }}(k)\) and \({\tilde{\pi }}(k) < {\tilde{\pi }}(j-1)\), so moving \({\tilde{\pi }}(k)\) to position j does not create a new ascent. On the other hand, if \(\ell =\ell '\), then \({\tilde{\pi }}(k)>{\tilde{\pi }}(j-1)\) but \({\tilde{\pi }}(j-1)-1\) must be to the left of \({\tilde{\pi }}(i)\). Therefore, also in this case, applying an iteration of \(\alpha \) will preserve the Fishburn condition.

We conclude that, if \(\pi \) is Fishburn, so is \(\alpha (\pi )\).

The reverse map \(\beta :{\mathscr {F}}_n(1243)\rightarrow {\mathscr {F}}_n(1423)\) is given by the following algorithm.

Algorithm\(\beta \): Let \(\tau \in {\mathscr {F}}_n(1243)\) and set \({\tilde{\tau }}=\tau \).
  1. Step 1:

    If \({\tilde{\tau }} \not \in \mathsf {Av}_n(1423)\), let \(i<j<k<\ell \) be the positions of the right-most 1423 pattern contained in \({\tilde{\tau }}\). Redefine \({\tilde{\tau }}\) by moving \({\tilde{\tau }}(j)\) to position k, shifting the entries at positions \(j+1\) through k one step to the left.

     
  2. Step 2:

    If \({\tilde{\tau }} \in \mathsf {Av}_n(1423)\), then return \(\beta (\tau )={\tilde{\tau }}\); otherwise go to Step 1.

     
In conclusion, the map \(\alpha \) gives a bijection \({\mathscr {F}}_n(1423)\rightarrow {\mathscr {F}}_n(1243)\).

With a similar argument, it can be verified that \(\alpha \) also maps \({\mathscr {F}}_n(1324)\rightarrow {\mathscr {F}}_n(1234)\) bijectively. Finally, the equivalence \({\mathscr {F}}_n(1243)\sim {\mathscr {F}}_n(1234)\) was shown in Theorem 3.5. \(\square \)

Theorem 3.7

\({\mathscr {F}}_n(3142)\sim {\mathscr {F}}_n(3124)\sim {\mathscr {F}}_n(1324)\).

Proof

We will define two maps
$$\begin{aligned} {\mathscr {F}}_n(3142) {\mathop {\longrightarrow }\limits ^{\alpha _1}} {\mathscr {F}}_n(3124) \;\text { and }\; {\mathscr {F}}_n(3124) {\mathop {\longrightarrow }\limits ^{\alpha _2}} {\mathscr {F}}_n(1324) \end{aligned}$$
through algorithms similar to the one introduced in the proof of Theorem 3.6.
Algorithm\(\alpha _1\): Let \(\pi \in {\mathscr {F}}_n(3142)\) and set \({\tilde{\pi }}=\pi \).
  1. Step 1:

    If \({\tilde{\pi }} \not \in \mathsf {Av}_n(3124)\), let \(i<j<k<\ell \) be the positions of the left-most 3124 pattern contained in \({\tilde{\pi }}\). Redefine \({\tilde{\pi }}\) by moving \({\tilde{\pi }}(\ell )\) to position k, shifting the entries at positions k through \(\ell -1\) one step to the right.

     
  2. Step 2:

    If \({\tilde{\pi }} \in \mathsf {Av}_n(3124)\), then return \(\alpha _1(\pi )={\tilde{\pi }}\); otherwise go to Step 1.

     
As \(\alpha \) in Theorem 3.6, the map \(\alpha _1\) is reversible and preserves the Fishburn condition. For an illustration of the latter claim, here is a sketch of a permutation \(\pi \in {\mathscr {F}}_n(3142)\) having a left-most 3124 pattern, together with the sketch of \({\tilde{\pi }}\) after one iteration of \(\alpha _1\):
where no elements of the permutation \(\pi \) may occur in the shaded regions.

Since \(\pi (k-1)<\pi (k)\) and \(\pi (\ell -1)<\pi (\ell )\), the Fishburn condition of \(\pi \) is preserved after the first iteration of \(\alpha _1\). Further, if \({\tilde{\pi }}\) has a left-most 3124 pattern with the third entry at position \(k'\), then we must have \(\ell >\ell '\) and \({\tilde{\pi }}(\ell )>{\tilde{\pi }}(i)\). If \({\tilde{\pi }}(\ell )<{\tilde{\pi }}(k'-1)\), no new ascent can be created when moving \({\tilde{\pi }}(\ell )\) to position \(k'\). Otherwise, if \({\tilde{\pi }}(\ell )>{\tilde{\pi }}(k'-1)\), then either \(\pi \) has ascents at the positions of these two entries or every entry between \({\tilde{\pi }}(\ell ')\) and \({\tilde{\pi }}(\ell )\) must be smaller than \({\tilde{\pi }}(j)\). Since \(\pi \in \mathsf {Av}_n(3142)\), the latter would imply that \({\tilde{\pi }}(k'-1)-1\) is to the left of \({\tilde{\pi }}(j)\). In any case, no pattern Open image in new window will be created in the next iteration of \(\alpha _1\).

Since any later iteration of \(\alpha _1\) may essentially be reduced to one of the above cases, we conclude that \(\alpha _1\) preserves the Fishburn condition.

Algorithm\(\alpha _2\): Let \(\pi \in {\mathscr {F}}_n(3124)\) and set \({\tilde{\pi }}=\pi \).
  1. Step 1:

    If \({\tilde{\pi }} \not \in \mathsf {Av}_n(1324)\), let \(i<j<k<\ell \) be the positions of the left-most 1324 pattern contained in \({\tilde{\pi }}\). Redefine \({\tilde{\pi }}\) by moving \({\tilde{\pi }}(j)\) to position i, shifting the entries at positions i through \(j-1\) one step to the right.

     
  2. Step 2:

    If \({\tilde{\pi }} \in \mathsf {Av}_n(1324)\), then return \(\alpha _2(\pi )={\tilde{\pi }}\); otherwise go to Step 1.

     
This map is reversible and preserves the Fishburn condition. As before, we will illustrate the Fishburn property by sketching the plot of a permutation \(\pi \in {\mathscr {F}}_n(3124)\) that contains a left-most 1324 pattern \(\pi (i)\), \(\pi (j)\), \(\pi (k)\), \(\pi (\ell )\), together with the sketch of the permutation \({\tilde{\pi }}\) obtained after one iteration of \(\alpha _2\):

Since no elements of the permutation \(\pi \) may occur in the shaded regions, we must have either \(i=1\) or \(\pi (i-1)>\pi (j)\). Consequently, moving \(\pi (j)\) to position i will not create a new ascent and the Fishburn condition will be preserved.

Similarly, if \({\tilde{\pi }}\) has a left-most 1324 pattern with first entry at a position different from \(i'\), or if \({\tilde{\pi }}(i)={\tilde{\pi }}(i')\) and \({\tilde{\pi }}(j)<{\tilde{\pi }}(i'-1)\), then no new ascent will be created and the next \({\tilde{\pi }}\) will be Fishburn. It is not possible to have \({\tilde{\pi }}(i)={\tilde{\pi }}(i')\) and \({\tilde{\pi }}(j)>{\tilde{\pi }}(i'-1)\).

In summary, \(\alpha _1\) and \(\alpha _2\) are both bijective maps. \(\square \)

The following theorem completes the enumeration of the Catalan class (see Table 4).

Theorem 3.8

\({\mathscr {F}}_n(3142)\sim {\mathscr {F}}_n(2143)\).

Proof

Let \(\gamma :{\mathscr {F}}_n(3142)\rightarrow {\mathscr {F}}_n(2143)\) be the map defined through the following algorithm.

Algorithm\(\gamma \): Let \(\pi \in {\mathscr {F}}_n(3142)\) and set \({\tilde{\pi }}=\pi \).
  1. Step 1:
    If \({\tilde{\pi }} \not \in \mathsf {Av}_n(2143)\), let \(i<j<k\) be the positions of the left-most 213 pattern contained in \({\tilde{\pi }}\) such that \({\tilde{\pi }}(i)\), \({\tilde{\pi }}(j)\), \({\tilde{\pi }}(k)\), \({\tilde{\pi }}(\ell )\) form a 2143 pattern for some \(\ell >k\). Let \(\ell _m\) be the position of the smallest such \({\tilde{\pi }}(\ell )\), and let
    $$\begin{aligned} Q=\{q\in [n]: {\tilde{\pi }}(i)\le {\tilde{\pi }}(q)<{\tilde{\pi }}(\ell _m)\}. \end{aligned}$$
    Redefine \({\tilde{\pi }}\) by replacing \({\tilde{\pi }}(\ell _m)\) with \({\tilde{\pi }}(i)\), adding 1 to \({\tilde{\pi }}(q)\) for every \(q\in Q\).
     
  2. Step 2:

    If \({\tilde{\pi }} \in \mathsf {Av}_n(2143)\), then return \(\gamma (\pi )={\tilde{\pi }}\); otherwise go to Step 1.

     
For example, if \(\pi =4312576\), then \(\gamma (\pi ) = 5412673\) (after 2 iterations, see Fig. 3).
Fig. 3

Algorithm \(\gamma \): \(4312576 \rightarrow 5312674 \rightarrow 5412673\)

The map \(\gamma \) is reversible. Moreover,
  1.    (a)

    since \({\tilde{\pi }}(\ell _m)\) is the smallest entry such that \({\tilde{\pi }}(i)<{\tilde{\pi }}(\ell _m)<{\tilde{\pi }}(k)\), replacing \({\tilde{\pi }}(\ell _m)\) with \({\tilde{\pi }}(i)\) (which is equivalent to moving the plot of \({\tilde{\pi }}(\ell _m)\) down to height \({\tilde{\pi }}(i)\)) will not create any new ascent at position \(\ell _m\);

     
  2.    (b)

    since \({\tilde{\pi }}(i)\) is chosen to be the first entry of a left-most 2143 pattern, \({\tilde{\pi }}(i)-1\) must be to the right of \({\tilde{\pi }}(i)\). Hence, replacing \({\tilde{\pi }}(i)\) by \({\tilde{\pi }}(i)+1\) cannot create a new pattern Open image in new window.

     
In conclusion, \(\gamma \) preserves the Fishburn condition and gives the claimed bijection. \(\square \)

4 Further Remarks

In this paper, we have discussed the enumeration of Fishburn permutations that avoid a pattern of size 3 or a pattern of size 4. In Sect. 2, we offer the complete picture for patterns of size 3, including the enumeration of indecomposable permutations.

Regarding patterns of size 4, we have proved the Wilf equivalence of eight permutation families counted by the Catalan numbers. We have also shown that \({\mathscr {F}}_n(1342)\) is enumerated by the binomial transform of the Catalan numbers. In general, there seems to be 13 Wilf equivalence classes of permutations that avoid a pattern of size 4, some of which appear to be in bijection with certain pattern avoiding ascent sequences ([7, A202061, A202062]). At this point in time, we do not know how the pattern avoidance of a Fishburn permutation is related to the pattern avoidance of an ascent sequence. It would be interesting to pursue this line of investigation.

Concerning indecomposable permutations, we leave the field open for future research. Note that Theorem 3.4 and Lemma 2.3 imply
$$\begin{aligned} |{\mathscr {F}}_n^{\textsf {ind}}(3142)| = C_{n-1}. \end{aligned}$$
The study of other patterns is unexplored territory, and our preliminary data suggests the existence of 19 Wilf equivalence classes listed in Table 5.

We are particularly curious about the class \({\mathscr {F}}_n^{\textsf {ind}}(2413)\) as it appears (based on limited data) to be equinumerous with the set \(\mathsf {Av}_{n-1}(2413,3412)\), cf. [7, A165546].

Footnotes

  1. 1.

    \(S_n\) denotes the set of permutations on \([n]=\{1,\dots ,n\}\).

  2. 2.

    This is a slightly different version of a bijection given by Krattenthaler [5].

Notes

References

  1. 1.
    Andrews, G.E., Sellers, J.A.: Congruences for the Fishburn numbers. J. Number Theory 161, 298–310 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bousquet-Mélou, M., Claesson, A., Dukes, M., Kitaev, S.: \((2+2)\)-free posets, ascent sequences and pattern avoiding permutations. J. Combin. Theory Ser. A 117(7), 884–909 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gao, A.L.L., Kitaev, S., Zhang, P.B.: On pattern avoiding indecomposable permutations. Integers 18, #A2 (2018)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Kitaev, S.: Patterns in Permutations and Words. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2011)Google Scholar
  5. 5.
    Krattenthaler, C.: Permutations with restricted patterns and Dyck paths. Adv. Appl. Math. 27(2-3), 510–530 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sapounakis, A., Tasoulas, I., Tsikouras, P.: Ordered trees and the inorder traversal. Discrete Math. 306(15), 1732–1741 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences. http://oeis.org
  8. 8.
    West, J.: Permutations with forbidden subsequences and stack-sortable permutations. Ph.D. Thesis. Massachusetts Institute of Technology, Cambridge (1990)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsPenn State AltoonaAltoonaUSA

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