## 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

- \(\triangleright \)
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}

*Fishburn permutations*. Further, we let \({\mathscr {F}}_n(\sigma )\) denote the class of Fishburn permutations in \({\mathscr {F}}_n\) that avoid the pattern \(\sigma \).

\(\sigma \)-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 |

\(\sigma \)-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 |

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, ... |

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

### Theorem 2.1

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

### Proof

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, ... |

*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

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.

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 \(1\ominus \pi \in {\mathscr {F}}_{n}(213)\).

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 (*n*, *n*) 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 \(j<i\).

*n*is defined as follows: Given \(\pi \in \mathsf {Av}_n(321)\), write

*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

*n*that avoid the subpath

*UUDU*. By [6, Proposition 5] we then have

### 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

*n*starts with

### Theorem 2.4

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

### Proof

### Theorem 2.5

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

### Proof

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

### Theorem 2.7

### 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 \(x\in [2,n-1]\).

*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

### Proof

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

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

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 \({\mathscr {A}}_n\subset {\mathscr {F}}_n(1342)\).

*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

### 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 \({\mathscr {F}}_n(231)\subseteq {\mathscr {F}}_n(3142)\).

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

- \(\circ \)
\(\pi (i)\) 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 \(\pi (k)<\pi (i)\).

### Theorem 3.5

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

### Proof

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 \({\tilde{\pi }}=\pi \).

- 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. - Step 2:
If \({\tilde{\pi }} \in \mathsf {Av}_n(1243)\), then return \(\alpha (\pi )={\tilde{\pi }}\); otherwise go to Step 1.

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

*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 \({\tilde{\tau }}=\tau \).

- 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. - Step 2:
If \({\tilde{\tau }} \in \mathsf {Av}_n(1423)\), then return \(\beta (\tau )={\tilde{\tau }}\); otherwise go to Step 1.

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

*Algorithm*\(\alpha _1\): Let \(\pi \in {\mathscr {F}}_n(3142)\) and set \({\tilde{\pi }}=\pi \).

- 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. - Step 2:
If \({\tilde{\pi }} \in \mathsf {Av}_n(3124)\), then return \(\alpha _1(\pi )={\tilde{\pi }}\); otherwise go to Step 1.

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 \({\tilde{\pi }}=\pi \).

- 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. - Step 2:
If \({\tilde{\pi }} \in \mathsf {Av}_n(1324)\), then return \(\alpha _2(\pi )={\tilde{\pi }}\); otherwise go to Step 1.

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 \({\tilde{\pi }}=\pi \).

- 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 letRedefine \({\tilde{\pi }}\) by replacing \({\tilde{\pi }}(\ell _m)\) with \({\tilde{\pi }}(i)\), adding 1 to \({\tilde{\pi }}(q)\) for every \(q\in Q\).$$\begin{aligned} Q=\{q\in [n]: {\tilde{\pi }}(i)\le {\tilde{\pi }}(q)<{\tilde{\pi }}(\ell _m)\}. \end{aligned}$$
- Step 2:
If \({\tilde{\pi }} \in \mathsf {Av}_n(2143)\), then return \(\gamma (\pi )={\tilde{\pi }}\); otherwise go to Step 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 \(\ell _m\);

- (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.

## 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.

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

## Notes

## References

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