Enumerating Permutations Avoiding More Than Three Babson-Steingrímsson Patterns

Antonio Bernini and Elisa Pergola Dipartimento di Sistemi e Informatica Università di Firenze viale G. B. Morgagni 65 50134 Firenze Italy

                     bernini@dsi.unifi.it
                      elisa@dsi.unifi.it

Abstract

Claesson and Mansour recently proposed some conjectures about the enumeration of the permutations avoiding more than three Babson-Steingrímsson patterns (generalized patterns of type (1, 2) or (2, 1)). The avoidance of one, two or three patterns has already been considered. Here, the cases of four and five forbidden patterns are solved and the exact enumeration of the permutations avoiding them is given, confirming the conjectures of Claesson and Mansour. The approach we use can be easily extended to the cases of more than five forbidden patterns.

1 Introduction

The results of the present paper concern the exact enumeration of the permutations, according to their length, avoiding any set of four or five generalized patterns of type (1, 2) or (2, 1) [2]. The cases of the permutations avoiding one, two or three generalized patterns (of the same types) were solved by Claesson [4], Claesson and Mansour [5], and Bernini, Ferrari and Pinzani [3], respectively. In particular, Claesson and Mansour [5] conjectured the plausible sequences enumerating the permutations of S_{n}(P), for any set P of three or more patterns.

Substantially, Bernini, Ferrari and Pinzani [3] conducted the proofs by finding the ECO construction [1] for the permutations avoiding three generalized patterns of type (1, 2) or (2, 1), encoding it with a succession rule and, finally, checking that this one leads to the enumerating sequence conjectured by Claesson and Mansour [5]. This approach could surely be used also for the investigation of the avoidance of four or five generalized patterns of type (1, 2) or (2, 1) and, maybe, it would allow to find some nice and interesting results: we think that, for instance, in some case new succession rules for known sequences would appear. However, this approach has one obstacle: the large number of cases to consider in order to exhaust all the conjectures. The line we are going to follow (see below) is simple and allows us to reduce the number of cases to be considered. Most of the results are summarized in several tables which are presented in Section 4. Really, the paper could appear an easy exercise, but we believe that it is a valuable contribute to the classification of permutations avoiding generalized patterns, started with Claesson, Mansour, Elizalde and Noy [6], Kitaev [8]. Moreover, it can be seen as the continuation of the work started by Bernini, Ferrari and Pinzani [3] for the fulfillment of the proofs of the conjectures presented by Claesson and Mansour [5].

1.1 Preliminaries

A (classical) pattern is a permutation σ ∈ S_{k} and a permutation π ∈ S_{n} avoids σ if there is no any subsequence π_{i_{1}}π_{i_{2}} … π_{i_{k}} with 1 ≤ i_{1} < i_{2} < … < i_{k} ≤ n which is order-isomorphic to σ. In other words, π must contain no subsequences having the entries in the same relative order of the entries of σ. Generalized patterns were introduced by Babson and Steingrímsson for the study of the mahonian statistics on permutations [2]. They are constructed by inserting one or more dashes among the elements of a classical pattern (two or more consecutive dashes are not allowed). For instance, 216 − 4 − 53 is a generalized pattern of length 6. The type (t_{1}, t_{2}, …, t_{h+1}) of a generalized pattern containing h dashes records the number of elements between two dashes (we suppose a dash at the beginning and at the end of the generalized pattern, but we omit it): the type of 216 − 4 − 53 is (3, 1, 2). A permutation π contains a generalized pattern τ if π contains τ in the classical sense and if any pair of elements of π corresponding to two adjacent elements of τ (not separated by a dash) are adjacent in π, too. For instance, π = 153426 contains 32−14 in the entries π_{2}π_{3}π_{5}π_{6} = 5326 or the pattern 3 − 214 in the entries π_{2}π_{4}π_{5}π_{6} = 5426. A permutation π avoids a generalized pattern τ if it does not contain τ. If P is a set of generalized patterns, we denote S_{n}(P) the permutations of length n of S (symmetric group) avoiding the patterns of P.

In the paper, we are interested in the generalized patterns of length three, which are of type (1, 2) or (2, 1). They are those ones specified in the set

\begin{aligned}M = & \{1 − 23, 12 − 3, 1 − 32, 13 − 2, 3 − 12, 31 − 2, 2 − 13, 21 − 3, \\ & 2 − 31, 23 − 1, 3 − 21, 32 − 1\}. \end{aligned}

In the sequel, sometimes we can refer to a generalized pattern of length three more concisely with pattern.

If π ∈ S, we define its reverse and its complement to be the permutations π^{r} and π^{c}, respectively, such that π_{i}^{r} = π_{n+1−i} and π_{i}^{c} = n + 1 − π_{i}. We generalize this definition to a generalized pattern τ obtaining its reverse τ^{r} by reading τ from right to left (regarding the dashes as particular entries) and its complement τ^{c} by considering the complement of τ regardless of the dashes which are left unchanged (e.g., if τ = 216 − 4 − 53, then τ^{r} = 35 − 4 − 612 and τ^{c} = 561 − 3 − 24). It is easy to check τ^{rc} = τ^{cr}. If P ⊆ M, the set {P, P^{r}, P^{c}, P^{rc}} is called the symmetry class of P (P^{r}, P^{c} and P^{rc} contain the reverses, the complements and the reverse-complements of the patterns specified in P, respectively). We have that |S_{n}(P)|=|S_{n}(P^{r})|=|S_{n}(P^{c})|=|S_{n}(P^{rc})| [9], therefore we can choose one of the four possible sets as the representative of a symmetry class, as far as the enumeration of S(R), R ∈ {P, P^{r}, P^{c}, P^{rc}}, is concerned.

1.2 The strategy

Looking at the table of the conjectures by Claesson and Mansour [5], it is possible to note that most of the sequences enumerating the permutations avoiding four patterns are the same of those ones enumerating the permutations avoiding three patterns. A similar fact happens when the forbidden patterns are four and five. This suggests to use the results for the case of three forbidden patterns (at our disposal) to deduce the proof of the conjectures for the case of four forbidden patterns and, similarly, use the results for the case of four forbidden patterns to solve the case of five forbidden patterns. Indeed, it is obvious that S(p_{1}, p_{2}, p_{3}, p_{4}) ⊆ S(p_{i_{1}}, p_{i_{2}}, p_{i_{3}}) (with i_{j} ∈ {1, 2, 3, 4} and p_{l} ∈ M). If the inverse inclusion can be proved for some patterns, then the classes S(p_{1}, p_{2}, p_{3}, p_{4}) and S(p_{i_{1}}, p_{i_{2}}, p_{i_{3}}) coincide and they are enumerated by the same sequence (a similar argument can be used for the cases of four and five forbidden patterns).

To this end, the following eight propositions are useful: each of them proves that if a permutation avoids certain patterns, then it avoids also a further pattern. Therefore, it is possible to apply one of them to a certain class S(p_{i_{1}}, p_{i_{2}}, p_{i_{3}}) to prove that S(p_{i_{1}}, p_{i_{2}}, p_{i_{3}}) ⊆ S(p_{1}, p_{2}, p_{3}, p_{4}) (the generalization to the case of four and five forbidden pattern is straightforward). The proof of the first one can be recovered in the paper of Claesson [4], where he proves that S(2 − 13) = S(213), and the three similar ones follow by symmetry.

Proposition 1.1. If π ∈ S(2 − 13), then π ∈ S(2 − 13, 21 − 3).

Proposition 1.2. If π ∈ S(31 − 2), then π ∈ S(31 − 2, 3 − 12).

Proposition 1.3. If π ∈ S(2 − 31), then π ∈ S(2 − 31, 23 − 1).

Proposition 1.4. If π ∈ S(13 − 2), then π ∈ S(13 − 2, 1 − 32).

Proposition 1.5. If π ∈ S(1 − 23, 2 − 13), then π ∈ S(1 − 23, 2 − 13, 12 − 3).

Proof. Suppose that π contains a 12 − 3 pattern in the entries π_{i}, π_{i+1} and π_{k} (k > i + 1). Let us consider the entry π_{i+2}. It can be neither π_{i+2} > π_{i+1} (since π_{i} π_{i+1} π_{i+2} would show a pattern 1 − 23) nor π_{i+2} < π_{i+1} (since π_{i+1} π_{i+2} π_{k} would show a pattern 21 − 3 which is forbidden thanks to Proposition 1.1). (The proof of the following proposition is very similar and is omitted.)

Proposition 1.6. If π ∈ S(1 − 23, 21 − 3), then π ∈ S(1 − 23, 21 − 3, 12 − 3).

Proposition 1.7. If π ∈ S(1 − 23, 2 − 31), then π ∈ S(1 − 23, 2 − 31, 12 − 3).

Proof. Suppose that a pattern 12 − 3 appear in π_{i}, π_{i+1} and π_{k}. If we consider the entry π_{k−1}, then it is easily seen that it can be neither π_{i} < π_{k−1} < π_{k} (the entries π_{i} π_{k−1} π_{k} would form a 1 − 23 pattern) nor π_{k−1} < π_{i} (the entries π_{i} π_{i+1} π_{k−1} would show a pattern 23 − 1 which is forbidden thanks to Proposition 1.3). Hence, π_{k−1} > π_{k}. We can repeat the same above argument for the entry π_{j}, j = k − 2, k − 3, …, i + 2, concluding each time that π_{j} > π_{j+1}. When j = i + 2 a pattern 1 − 23 is shown in π_{i} π_{i+1} π_{i+2}, which is forbidden.

Proposition 1.8. If π ∈ S(1 − 23, 23 − 1), then π ∈ S(1 − 23, 23 − 1, 12 − 3).

This last proposition can be be proved by simply adapting the argument of the proof of the preceding one.

2 Permutations avoiding four patterns

First of all we recall the results relating to the case of three forbidden patterns [3] in Tables 1 and 2. For the sake of brevity, for each symmetry class only a representative is reported. In the first column of these tables, a name to each symmetry class is given, the second one shows the three forbidden patterns (the representative) and the third one indicates the sequence enumerating the permutations avoiding the specified patterns.

Having at our disposal the results for the permutations avoiding three patterns, the proofs for the case of four forbidden patterns are conducted following the line indicated in the previous section. These proofs are all summarized in tables. Tables 3, 4 and 5 are related to the permutations avoiding four patterns enumerated by the sequences {n}_{n≥1}, {F_{n}}_{n≥1} and {2^{n−1}}_{n≥1}, respectively (the sequence {F_{n}}_{n≥1} denotes the Fibonacci numbers). The empty permutation with length n = 0 is not considered, therefore the length is n ≥ 1. The Tables have to be read as follows: consider the representative of the symmetry class specified in the rightmost column of each table; apply the proposition indicated in the preceding column to the three forbidden patterns which one can find in Tables 1 and 2 to obtain the four forbidden patterns written in the column named avoided patterns. At this point, as we explained in the previous section, the permutations avoiding these four patterns are enumerated by the same sequence enumerating the permutations avoiding the three patterns contained in the representative of the symmetry class indicated in the rightmost column.

The first column of Table 3 and 4 specifies a name for the the symmetry class represented by the four forbidden patterns of the second column. This name is useful in the next section. Table 6 indicates in the first column the sequence enumerating the permutations avoiding the patterns of the second column, which are obtained as in the above tables.

2.1 Classes enumerated by {0}_{n≥k}.

The classes of four patterns avoiding permutations enumerated by the ultimately constant sequence {0}_{n≥k} can be handled in a very simple way (the value of k ∈ ℕ depends on the considered patterns, however it is never greater than 4). If S(q_{1}, q_{2}, q_{3}), q_{i} ∈ M, is a class of permutations avoiding three patterns such that |S_{n}(q_{1}, q_{2}, q_{3})| = 0, for n ≥ k, then it is easily seen that S(q_{1}, q_{2}, q_{3}, r), ∀r ∈ M, is also enumerated by the same sequence. Then, each symmetry class from C1 to C7 (see Table 2) generates nine symmetry classes by choosing the pattern r ≠ q_{i}, i = 1, 2, 3. It is easy to see that some of the classes we obtain in this way are equal, thanks to the operations of reverse, complement and reverse-complement. In Table 7, only the different possible cases are presented. Here, the four forbidden patterns are recovered by adding a pattern of a box of the second column to the three patterns specified in the box to its right at the same level (rightmost column). The representative so obtained is recorded in the leftmost column with a name, which will be useful in the next section.

2.2 Classes enumerated by {2}_{n≥2}.

The enumerating sequences encountered till now (see Tables 3, 4, 5, 6, 7) are all involved in the enumeration of some class of permutations avoiding three patterns (Tables 1, 2). Therefore, applying the eight propositions of the previous section to the classes of Table 1 and 2, the three forbidden patterns have been increased by one pattern, obtaining Table 3, 4, 5, 6 and 7. For the classes enumerated by the sequence {2}_{n≥2} it is not possible to use the same strategy, since there are no classes of permutations avoiding three patterns enumerated by that sequence. The proofs, in this case, use four easy propositions whose proofs can be directly derived from the statement of the first four propositions of the Introduction. We prefer to explicit them the same.

Proposition 2.1. If a permutation π contains the pattern 23−1, then it contains the pattern 2 − 31, too.

Taking the reverse, the complement and the reverse-complement of the patterns involved in Prop. 2.1, the following propositions are obtained:

Proposition 2.2. If a permutation π contains the pattern 1−32, then it contains the pattern 13 − 2, too.

Proposition 2.3. If a permutation π contains the pattern 21−3, then it contains the pattern 2 − 13, too.

Proposition 2.4. If a permutation π contains the pattern 3−12, then it contains the pattern 31 − 2, too.

In Table 8 the results relating to the enumeration of the permutations avoiding four patterns enumerated by the sequence {2}_{n≥2} (whose proofs are contained in the six next propositions) are summarized. The four forbidden patterns can be recovered by choosing one pattern from each column, in the same box-row of the table.

In the sequel, p_{i} ∈ A_{i} with i = 1, 2, 3, 4 where A_{i} is a subset of generalized patterns.

Proposition 2.5. Let A_{1} = {1 − 23}, A_{2} = {2 − 31, 23 − 1}, A_{3} = {1 − 32, 13 − 2} and A_{4} = {3 − 12, 31 − 2}. Then |S_{n}(p_{1}, p_{2}, p_{3}, p_{4})| = 2 and S_{n} = {n (n − 1) … 3 2 1, (n − 1) (n − 2) … 3 2 1 n}.

Proof. Let σ ∈ S_{n}(p_{2}, p_{3}). Then, σ_{1} = n or σ_{n} = n, otherwise, if σ_{i} = n with i ≠ 1, n , the entries σ_{i−1}σ_{i}σ_{i+1} would be a forbidden pattern p_{2} or p_{3}.

If ρ ∈ S_{n}(p_{1}, p_{3}), then ρ_{n−1} = 1 or ρ_{n} = 1, otherwise, if ρ_{i} = 1 with i < n − 1, then the entries ρ_{i}ρ_{i+1}ρ_{i+2}, would be a forbidden pattern p_{1} or p_{3}.

Therefore, if π ∈ S_{n}(p_{1}, p_{2}, p_{3}), then there are only the following three cases for π:

1. π_{n} = n and π_{n−1} = 1. In this case π = (n − 1) (n − 2) … 2 1 n, otherwise, if an ascent appears in π_{j}π_{j+1} with j = 1, 2, …, n − 3, the entries π_{j}π_{j+1}π_{n−1} would show the pattern 23 − 1 and π would contain the pattern 2 − 31, too (see Prop. 2.1).

2. π_{1} = n and π_{n} = 1. In this case π = n (n − 1) … 3 2 1, otherwise, if an ascent appears in π_{j}π_{j+1} with j = 2, 3, …, n − 2, the entries π_{j}π_{j+1}π_{n} would show the pattern 23 − 1 and π would contain the pattern 2 − 31, too (see Prop. 2.1).

3. π_{1} = n and π_{n−1} = 1 (and π_{n} = k < n).

If π has to avoid the pattern p_{4}, too (π ∈ S_{n}(p_{1}, p_{2}, p_{3}, p_{4})), then the third above case is not allowed since π_{1}π_{n−1}π_{n} are a 3 − 12 pattern which induces an occurrence of 31 − 2 in π (Prop. 2.4).

Proposition 2.6. Let A_{1} = {1 − 23}, A_{2} = {2 − 13, 21 − 3}, A_{3} = {1 − 32, 13 − 2} and A_{4} = {3 − 12, 31 − 2}. Then |S_{n}(p_{1}, p_{2}, p_{3}, p_{4})| = 2 and S_{n} = {n (n − 1) … 3 2 1, (n − 1) n (n − 2) (n − 3) … 2 1}.

Proof. If σ ∈ S_{n}(p_{1}, p_{2}), then π_{1} = n or π_{2} = n. If ρ ∈ S_{n}(p_{1}, p_{3}), then π_{n} = 1 or π_{n−1} = 1. Then, if π ∈ S_{n}(p_{1}, p_{2}, p_{3}), there are only the four following cases:

1. π_{1} = n and π_{n} = 1.

2. π_{2} = n and π_{n} = 1. In this case π_{1} = n − 1, otherwise if π_{k} = n − 1 with k > 3, then π_{k−2}π_{k−1}π_{k} is a 1−23 pattern or a 21−3 pattern which induces an occurrence of 2−13 (Prop. 2.3). If k = 3, then π_{1}π_{2}π_{3} is a 1 − 32 or 13 − 2 pattern which are forbidden.

3. π_{1} = n and π_{n−1} = 1.

4. π_{2} = n and π_{n−1} = 1. For the same reasons of case 2, it is π_{1} = n − 1.

If π has to avoid p_{4}, too (π ∈ S_{n}(p_{1}, p_{2}, p_{3}, p_{4})), then the third and the fourth above cases are not allowed since π_{1}π_{n−1}π_{n} are a 3 − 12 pattern which induces an occurrence of 31 − 2 (Prop. 2.4). Moreover, the permutations of the above cases 1 and 2, must be such that there are not ascents π_{i}π_{i+1} between n and 1 in order to avoid p_{4}. Then, π = n (n − 1) … 3 2 1 or π = (n − 1) n (n − 2) … 3 2 1.

Proposition 2.7. Let A_{1} = {2−13, 21−3}, A_{2} = {2−31, 23−1}, A_{3} = {1−32, 13−2} and A_{4} = {3 − 12, 31 − 2}. Then |S_{n}(p_{1}, p_{2}, p_{3}, p_{4})| = 2 and S_{n} = {n (n − 1) … 2 1, 1 2 … n}.

Proof. It is easily seen that each three consecutive elements of π can only be in increasing or decreasing order.

Proposition 2.8. Let A_{1} = {12 − 3}, A_{2} = {2 − 13, 21 − 3}, A_{3} = {2 − 31, 23 − 1} and A_{4} = {32 − 1}. Then |S_{n}(p_{1}, p_{2}, p_{3}, p_{4})| = 2 and S_{n} = {1 n 2 (n − 1) …, n 1 (n − 1) 2 …}.

Proof. If π ∈ S_{n}(p_{1}, p_{2}, p_{3}, p_{4}), then it is easy to see that π_{1}π_{2} = 1 n or π_{1}π_{2} = n 1. Considering the sub-permutation π_{2}π_{3} … π_{n}, in the same way we deduce π_{2}π_{3} = 2 (n − 1) or π_{2}π_{3} = (n − 1) 2. The thesis follows by recursively using the above argument.

Proposition 2.9. Let A_{1} = {1 − 23}, A_{2} = {2 − 13, 21 − 3}, A_{3} = {2 − 31, 23 − 1} and A_{4} = {3 − 12, 31 − 2}. Then |S_{n}(p_{1}, p_{2}, p_{3}, p_{4})| = 2 and S_{n} = {n (n − 1) … 1, 1 n (n − 1) … 3 2}.

Proof. Let π ∈ S_{n}(p_{1}, p_{2}, p_{3}, p_{4}). It is π_{1} = n or π_{2} = n, otherwise a 1 − 23 or p_{2} pattern would appear.

If π_{1} = n, then π = n (n−1) … 1 since if an ascent appears in π_{i}π_{i+1}, the entries π_{1}π_{i}π_{i+1} are a p_{4} pattern.

If π_{2} = n, then π_{1} = 1 since the p_{3} pattern has to be avoided. Moreover, in this case, it is π_{j} > π_{j+1} with j = 3, 4, …, (n−1) in order to avoid 1−23. Then π = 1 n (n−1) … 2 1.

Proposition 2.10. Let A_{1} = {1−23}, A_{2} = {2−13, 21−3}, A_{3} = {2−31, 23−1} and A_{4} = {1−32, 13−2}. Then |S_{n}(p_{1}, p_{2}, p_{3}, p_{4})| = 2 and S_{n} = {n (n−1) … 3 2 1, n (n−1) … 3 1 2}.

Proof. Let π ∈ S_{n}(p_{1}, p_{2}, p_{3}, p_{4}). The entries 1 and 2 have to be adjacent in order to avoid p_{3} and p_{4} and π_{n} = 1 or π_{n−1} = 1 in order to avoid p_{1} and p_{4}. So, π_{n−1}π_{n} = 1 2 or π_{n−1}π_{n} = 2 1. Moreover, each couple of adjacent elements π_{j}π_{j+1} must be a descent, otherwise a 23 − 1 pattern (which induces an occurrences of 2 − 31) would appear. Then π = n (n − 1) … 3 2 1 or π = n (n − 1) … 3 1 2.

The conjecture stated by Claesson and Mansour [5] about the permutations enumerated by {2}_{n≥2} declares that there are 42 symmetry classes of such permutations, while from Table 8 it is possible to deduce 52 symmetry classes. However, some of them represent the same class: for example the symmetry class {2 − 13, 2 − 31, 1 − 32, 31 − 2} is the same of {2 − 13, 23 − 1, 13 − 2, 31 − 2} (the second one is the reverse of the first one). Note that the repetitions come out only from the third box-row of Table 8.

3 Permutations avoiding five patterns

3.1 Classes enumerated by {1}_{n≥1}

The sequence {1}_{n≥1} does not enumerate any class of permutations avoiding four patterns, so that we can not apply the same method of the previous section using the propositions of the Introduction.

Referring to Proposition 2.7, we deduce that there are sixteen different classes S_{n}(p_{1}, p_{2}, p_{3}, p_{4}) such that p_{i} ∈ A_{i} with i = 1, 2, 3, 4. We recall that |S_{n}(p_{1}, p_{2}, p_{3}, p_{4})| = 2 and S_{n}(p_{1}, p_{2}, p_{3}, p_{4}) = {n (n − 1) … 2 1, 1 2 … n}. If a permutation π ∈ S_{n}(p_{1}, p_{2}, p_{3}, p_{4}) has to avoid the pattern 1 − 23, too, then π = n (n − 1) … 2 1 and |S_{n}(p_{1}, p_{2}, p_{3}, p_{4}, 1 − 23)| = 1. Then, the five forbidden patterns avoided by the permutations enumerated by {1}_{n≥1} can be recovered by considering the four patterns chosen from the third box-row of Table 8 (one pattern from each column) and the pattern 1 − 23. We do not present the corresponding table.

3.2 Classes enumerated by {0}_{n≥4}

This case is treated as the case of the permutations avoiding four patterns. It is sufficient to add a pattern r ∈ M to each representative (from O1 to O37 in Table 7) of four forbidden patterns of Table 7 in order to obtain a representative T of five forbidden patterns such that |S_{n}(T)| = 0, n ≥ 4. In Table 9 we present the different representatives T which can be derived from Table 7. The five forbidden patterns of each representative are a pattern chosen in a box of the first column and the four patterns indicated by the representative (which refer to Table 7) in the second box at the same level. In the table, only the different representatives of five patterns are presented.

3.3 Classes enumerated by {2}_{n≥2}, {n}_{n≥1}, {F_{n}}_{n≥1}

Tables 10 and 11 summarize the results related to the permutations avoiding five patterns enumerated by {2}_{n≥2}. The five forbidden patterns are obtained by considering a representative of four forbidden patterns of the rightmost column and the pattern specified in the corresponding box of the preceding column. The first column indicates which is the proposition to apply. Note that each representative of four patterns (rightmost column) can be found in Table 8.

The reading of Tables 12 and 13 (related to the sequences {n}_{n≥1} and {F_{n}}_{n≥1}, respectively) is as usual: apply the proposition specified in the first column to recover the representative of five forbidden patterns which is composed by the pattern of the second column and the four patterns of the representative indicated in the rightmost column. Here, the names of the representatives refer to Tables 3 and 4.

4 Tables

symmetry class avoided patterns enumerating sequence
N1  {1-23,2-13,3-12}
N2  {1-23,2-13,31-2}
N3  {1-23,21-3,3-12}
N4  {1-23,21-3,31-2}
N5  {12-3,3-12,2-13}
N6  {12-3,3-12,21-3}
N7  {12-3,31-2,2-13}
N8  {12-3,31-2,21-3}
N9  {1-23,2-13,2-31}
N10  {1-23,2-13,23-1}
N11  {1-23,21-3,2-31}
N12  {1-23,21-3,23-1}
N13 {2-13,2-31,1-32} {n}_{n≥1}
N14  {2-13,23-1,1-32}
N15  {2-13,2-31,13-2}
N16  {2-13,23-1,13-2}
N17  {2-31,21-3,13-2}
N18  {2-31,21-3,1-32}
N19  {13-2,21-3,23-1}
N20  {21-3,23-1,1-32}
N21  {1-23,2-31,31-2}
N22  {1-23,23-1,31-2}
N23  {1-23,2-31,3-12}
N24  {1-23,1-32,3-21}
A1  {1-23,12-3,23-1}
A2  {2-31,23-1,1-32}
A3  {2-31,23-1,13-2}
A4  {1-23,12-3,2-13}
A5 {1-23,2-13,21-3} {2^{n−1}}_{n≥1}
A6  {1-23,3-12,31-2}
A7  {31-2,3-12,13-2}
A8  {31-2,3-12,1-32}
A9  {2-13,21-3,1-32}
A10  {2-13,21-3,13-2}
A11 {1-23,23-1,3-12} {2n − 2 + 1}_{n≥1}

Table 1: permutations avoiding three patterns

symmetry class  avoided patterns  enumerating sequence
F1  {1 − 23, 2 − 13, 1 − 32}
F2  {1 − 23, 2 − 13, 13 − 2}
F3  {1 − 23, 21 − 3, 13 − 2}
F4 {1 − 23, 13 − 2, 3 − 12} {F_{n}}_{n≥1}
F5  {1 − 23, 1 − 32, 3 − 12}
F6  {1 − 23, 1 − 32, 31 − 2}
F7  {1 − 23, 13 − 2, 31 − 2}
M1  {1 − 23, 12 − 3, 21 − 3}
M2  {12 − 3, 21 − 3, 2 − 13}  {M_{n}}_{n≥1}
B1  \{1 − 23, 21 − 3, 1 − 32\}  \{\binom{n}{⌈n/2⌉}\}_{n≥1}
B2  {12 − 3, 1 − 23, 31 − 2}
B3  \{1 − 23, 2 − 31, 23 − 1\}  \{1 + \binom{n}{2}\}
C8  {12 − 3, 2 − 13, 32 − 1}  {3}_{n≥3}
C1  {1 − 23, 2 − 13, 3 − 21}
C2  {1 − 23, 23 − 1, 32 − 1}
C3  {1 − 23, 2 − 13, 32 − 1}
C4 {1 − 23, 12 − 3, 3 − 21} {0}_{n≥k}
C5  {1 − 23, 21 − 3, 3 − 21}
C6  {1 − 23, 21 − 3, 32 − 1}
C7  {1 − 23, 2 − 31, 32 − 1}

Table 2: permutations avoiding three patterns

Enumerating sequence: {n}_{n≥1}

name  avoided patterns  apply Proposition to the symmetry class
d1 {1 − 23, 2 − 13, 3 − 12, 21 − 3}  1.1  N1
d2 {1 − 23, 2 − 13, 31 − 2, 21 − 3}  1.1  N2
d3 {1 − 23, 2 − 13, 31 − 2, 3 − 12}  1.2  N2
d4 {1 − 23, 21 − 3, 31 − 2, 3 − 12}  1.2  N4
d5 {12 − 3, 3 − 12, 2 − 13, 21 − 3}  1.1  N5
d6 {12 − 3, 31 − 2, 2 − 13, 21 − 3}  1.1  N7
d7 {12 − 3, 31 − 2, 2 − 13, 3 − 12}  1.2  N7
d8 {12 − 3, 31 − 2, 21 − 3, 3 − 12}  1.2  N8
d9 {1 − 23, 2 − 13, 2 − 31, 21 − 3}  1.1  N9
d10 {1 − 23, 2 − 13, 2 − 31, 23 − 1}  1.3  N9
d11 {1 − 23, 2 − 13, 23 − 1, 21 − 3}  1.1  N10
d12 {1 − 23, 21 − 3, 2 − 31, 23 − 1}  1.3  N11
d13 {2 − 13, 2 − 31, 1 − 32, 21 − 3}  1.1  N13
d14 {2 − 13, 2 − 31, 1 − 32, 23 − 1}  1.3  N13
d15 {2 − 13, 23 − 1, 1 − 32, 21 − 3}  1.1  N14
d16 {2 − 13, 2 − 31, 13 − 2, 21 − 3}  1.1  N15
d17 {2 − 13, 2 − 31, 13 − 2, 23 − 1}  1.3  N15
d18 {2 − 13, 2 − 31, 13 − 2, 1 − 32}  1.4  N15
d19 {2 − 13, 23 − 1, 13 − 2, 21 − 3}  1.1  N16
d20 {2 − 13, 23 − 1, 13 − 2, 1 − 32}  1.4  N16
d21 {2 − 31, 21 − 3, 13 − 2, 23 − 1}  1.3  N17
d22 {2 − 31, 21 − 3, 13 − 2, 1 − 32}  1.4  N17
d23 {2 − 31, 21 − 3, 1 − 32, 23 − 1}  1.3  N18
d24 {13 − 2, 21 − 3, 23 − 1, 1 − 32}  1.4  N19
d25 {1 − 23, 2 − 31, 31 − 2, 23 − 1}  1.3  N21
d26 {1 − 23, 2 − 31, 31 − 2, 3 − 12}  1.2  N21
d27 {1 − 23, 23 − 1, 31 − 2, 3 − 12}  1.2  N22
d28 {1 − 23, 2 − 31, 3 − 12, 23 − 1}  1.3  N23
d29 {1 − 23, 2 − 13, 31 − 2, 12 − 3}  1.5  N2
d30 {1 − 23, 2 − 13, 3 − 12, 12 − 3}  1.5  N1
d31 {1 − 23, 2 − 13, 2 − 31, 12 − 3}  1.5  N9
d32 {1 − 23, 2 − 13, 23 − 1, 12 − 3}  1.5  N10
d33 {1 − 23, 21 − 3, 2 − 31, 12 − 3}  1.6  N11
d34 {1 − 23, 21 − 3, 23 − 1, 12 − 3}  1.6  N12
d35 {1 − 23, 21 − 3, 31 − 2, 12 − 3}  1.6  N4
d36 {1 − 23, 21 − 3, 3 − 12, 12 − 3}  1.6  N3
d37 {1 − 23, 2 − 31, 3 − 12, 12 − 3}  1.7  N23
d38 {1 − 23, 2 − 31, 31 − 2, 12 − 3}  1.7  N21

Table 3: permutations avoiding four patterns

Enumerating sequence: {F_{n}}_{n≥1}
name  avoided patterns  apply Proposition to the symmetry class
e1 {1 − 23, 2 − 13, 1 − 32, 21 − 3}  1.1  F1
e2 {1 − 23, 2 − 13, 1 − 32, 12 − 3}  1.5  F1
e3 {1 − 23, 2 − 13, 13 − 2, 21 − 3}  1.1  F2
e4 {1 − 23, 2 − 13, 13 − 2, 1 − 32}  1.4  F2
e5 {1 − 23, 2 − 13, 13 − 2, 12 − 3}  1.5  F2
e6 {1 − 23, 21 − 3, 13 − 2, 1 − 32}  1.4  F3
e7 {1 − 23, 13 − 2, 3 − 12, 1 − 32}  1.4  F4
e8 {1 − 23, 1 − 32, 31 − 2, 3 − 12}  1.2  F6
e9 {1 − 23, 13 − 2, 31 − 2, 1 − 32}  1.4  F7
e10 {1 − 23, 13 − 2, 31 − 2, 3 − 12}  1.2  F7

Table 4: permutations avoiding four patterns Enumerating sequence: {2^{n−1}}_{n≥1}

avoided patterns  apply Proposition to the symmetry class
{1 − 23, 12 − 3, 2 − 13, 21 − 3}  1.1  A4
{31 − 2, 3 − 12, 13 − 2, 1 − 32}  1.4  A7
{2 − 13, 21 − 3, 13 − 2, 1 − 32}  1.4  A10
{2 − 31, 23 − 1, 1 − 32, 13 − 2}  1.4  A3

Table 5: permutations avoiding four patterns

Enumerating sequence

avoided patterns apply Proposition to the symmetry class

\{1 + \binom{n}{2}\}_{n≥1}  \{12 − 3, 1 − 23, 31 − 2, 3 − 12\}  1.2  B2
\{\binom{n}{⌈n/2⌉}\}_{n≥1}  \{1 − 23, 21 − 3, 1 − 32, 12 − 3\}  1.6  B1
{2^{n−2} + 1}_{n≥1} {1 − 23, 23 − 1, 3 − 12, 12 − 3}  1.8  A11
{3}_{n≥3}  {12 − 3, 2 − 13, 32 − 1, 21 − 3}  1.1  C8

Table 6: permutations avoiding four patterns

Enumerating sequence: {0}_{n≥k}

name choose a pattern from the following to add  to the symmetry class
O1  12 − 3
O2  1 − 32
O3  13 − 2
O4  3 − 12
O5 31 − 2 {1 − 23, 2 − 13, 3 − 21} (C1)
O6  21 − 3
O7  2 − 31
O8  23 − 1
O9  32 − 1
O10  12 − 3
O11  1 − 32
O12  13 − 2
O13  3 − 12
O14 31 − 2 {1 − 23, 23 − 1, 32 − 1} (C2)
O15  2 − 13
O16  21 − 3
O17  2 − 31
O18  3 − 21
O19  12 − 3
O20  13 − 2
O21  3 − 12
O22 31 − 2 {1 − 23, 2 − 13, 32 − 1} (C3)
O23  21 − 3
O24  2 − 31
O25  31 − 2
O26  1 − 32
O27 23 − 1 {1 − 23, 12 − 3, 3 − 21} (C4)
O28  32 − 1
O29  1 − 32
O30  13 − 2
O31 3 − 12 {1 − 23, 21 − 3, 3 − 21} (C5)
O32  31 − 2
O33  23 − 1
O34  13 − 2
O35 3 − 12 {1 − 23, 21 − 3, 32 − 1} (C6)
O36  2 − 31
O37 13 − 2 {1 − 23, 2 − 31, 32 − 1} (C7)

Table 7: permutations avoiding four patterns

Enumerating sequence: {2}_{n≥2}

1st pattern 2nd pattern 3rd pattern 4th pattern
1 − 23  \begin{matrix}2 − 31 or \\ 23 − 1 \end{matrix} \begin{matrix}1 − 32 or \\ 13 − 2 \end{matrix} \begin{matrix}3 − 12 or \\ 31 − 2 \end{matrix}
1 − 23  \begin{matrix}2 − 13 or \\ 21 − 3 \end{matrix} \begin{matrix}1 − 32 or \\ 13 − 2 \end{matrix} \begin{matrix}3 − 12 or \\ 31 − 2 \end{matrix}
\begin{matrix}2 − 13 or \\ 21 − 3 \end{matrix} \begin{matrix}2 − 31 or \\ 23 − 1 \end{matrix} \begin{matrix}1 − 32 or \\ 13 − 2 \end{matrix} \begin{matrix}3 − 12 or \\ 31 − 2 \end{matrix}
12 − 3  \begin{matrix}2 − 13 or \\ 21 − 3 \end{matrix} \begin{matrix}2 − 31 or \\ 23 − 1 \end{matrix}  32 − 1
1 − 23  \begin{matrix}2 − 13 or \\ 21 − 3 \end{matrix} \begin{matrix}2 − 31 or \\ 23 − 1 \end{matrix} \begin{matrix}3 − 12 or \\ 31 − 2 \end{matrix}
1 − 23  \begin{matrix}2 − 13 or \\ 21 − 3 \end{matrix} \begin{matrix}2 − 31 or \\ 23 − 1 \end{matrix} \begin{matrix}1 − 32 or \\ 13 − 2 \end{matrix}

Table 8: permutations avoiding four patterns

Enumerating sequence: {0}_{n≥k}

choose a pattern from the following to add to the symmetry class
\begin{matrix}21 − 3, 2 − 31, 23 − 1, 1 − 32, 13 − 2, \\ 3 − 12, 31 − 2, 32 − 1 \end{matrix}  O1
\begin{matrix}2 − 31, 23 − 1, 1 − 32, 13 − 2, 3 − 12, \\ 31 − 2 \end{matrix}  O6
\begin{matrix}21 − 3, 2 − 31, 23 − 1, 1 − 32, 3 − 12, \\ 31 − 2 \end{matrix}  O19
\begin{matrix}2 − 31, 23 − 1, 1 − 32, 13 − 2, 3 − 12 \\ 31 − 2 \end{matrix}  O23
1 − 32, 13 − 2, 3 − 12, 31 − 2, 32 − 1  O8
23 − 1, 1 − 32, 13 − 2, 31 − 2, 3 − 12  O24
12 − 3, 32 − 1, 13 − 2, 3 − 12, 31 − 2  O29
3 − 12, 13 − 2, 1 − 32, 23 − 1, 12 − 3  O36
1 − 32, 13 − 2, 3 − 12, 31 − 2  O15
13 − 2, 3 − 12, 31 − 2  O2
1 − 32, 2 − 31, 31 − 2  O10
12 − 3, 13 − 2, 3 − 12  O32
1 − 32, 13 − 2, 32 − 1  O33
3 − 21, 23 − 1, 1 − 32  O34
3 − 12, 31 − 2  O3
1 − 32, 13 − 2  O7
13 − 2, 3 − 12  O9
21 − 3, 13 − 2  O11
1 − 32, 3 − 12  O20
3 − 12, 23 − 1  O26
2 − 31, 32 − 1  O27
3 − 12  O5
2 − 31  O12
1 − 32  O21
3 − 12  O30
23 − 1  O35

Table 9: permutations avoiding five patterns

Enumerating sequence: {2}_{n≥2}

thanks to Proposition add the pattern to the patterns
1.1  21 − 3  \begin{matrix}\{1 − 23, 2 − 13, 1 − 32, 3 − 12\} or \\ \{1 − 23, 2 − 13, 1 − 32, 31 − 2\} or \\ \{1 − 23, 2 − 13, 13 − 2, 3 − 12\} or \\ \{1 − 23, 2 − 13, 1 − 32, 31 − 2\} \end{matrix}
1.1  21 − 3  \begin{matrix}\{1 − 23, 2 − 13, 2 − 31, 1 − 32\} or \\ \{1 − 23, 2 − 13, 2 − 31, 13 − 2\} or \\ \{1 − 23, 2 − 13, 23 − 1, 1 − 32\} or \\ \{1 − 23, 2 − 13, 23 − 1, 13 − 2\} \end{matrix}
1.1  21 − 3  \begin{matrix}\{1 − 23, 2 − 13, 2 − 31, 3 − 12\} or \\ \{1 − 23, 2 − 13, 2 − 31, 31 − 2\} or \\ \{1 − 23, 2 − 13, 23 − 1, 3 − 12\} or \\ \{1 − 23, 2 − 13, 23 − 1, 31 − 2\} \end{matrix}
1.1  21 − 3  \begin{matrix}\{1 − 23, 2 − 13, 23 − 1, 32 − 1\} or \\ \{1 − 23, 2 − 13, 2 − 31, 32 − 1\} \end{matrix}
1.1  21 − 3  \begin{matrix}\{2 − 13, 2 − 31, 1 − 32, 3 − 12\} or \\ \{2 − 13, 2 − 31, 1 − 32, 31 − 2\} or \\ \{2 − 13, 2 − 31, 13 − 2, 31 − 2\} \end{matrix}
1.1  21 − 3  \begin{matrix}\{2 − 13, 23 − 1, 1 − 32, 31 − 2\} or \\ \{2 − 13, 23 − 1, 1 − 32, 3 − 12\} or \\ \{2 − 13, 23 − 1, 13 − 2, 31 − 2\} or \\ \{2 − 13, 23 − 1, 13 − 2, 3 − 12\} \end{matrix}
1.2  3 − 12  \begin{matrix}\{1 − 23, 2 − 13, 2 − 31, 31 − 2\} or \\ \{1 − 23, 2 − 13, 23 − 1, 31 − 2\} \end{matrix}
1.2  3 − 12  \begin{matrix}\{1 − 23, 2 − 13, 1 − 32, 31 − 2\} or \\ \{1 − 23, 2 − 13, 13 − 2, 31 − 2\} \end{matrix}
1.2  3 − 12  \begin{matrix}\{1 − 23, 2 − 13, 2 − 31, 31 − 2\} or \\ \{1 − 23, 2 − 13, 23 − 1, 31 − 2\} \end{matrix}
1.2  3 − 12  \begin{matrix}\{1 − 23, 21 − 3, 1 − 32, 31 − 2\} or \\ \{1 − 23, 21 − 3, 13 − 2, 31 − 2\} \end{matrix}
1.2  3 − 12  \begin{matrix}\{1 − 23, 2 − 31, 1 − 32, 31 − 2\} or \\ \{1 − 23, 2 − 31, 13 − 2, 31 − 2\} \end{matrix}
1.2  3 − 12  \begin{matrix}\{1 − 23, 23 − 1, 1 − 32, 31 − 2\} or \\ \{1 − 23, 23 − 1, 13 − 2, 31 − 2\} \end{matrix}
1.3  23 − 1  {2 − 13, 2 − 31, 1 − 32, 31 − 2}
1.3  23 − 1  \begin{matrix}\{1 − 23, 2 − 31, 13 − 2, 3 − 12\} or \\ \{1 − 23, 2 − 31, 13 − 2, 31 − 2\} \end{matrix}

Table 10: permutations avoiding five patterns

Enumerating sequence: {2}_{n≥2}

thanks to Proposition add the pattern to the patterns
1.3  23 − 1  \begin{matrix}\{1 − 23, 2 − 31, 1 − 32, 3 − 12\} or \\ \{1 − 23, 2 − 31, 1 − 32, 31 − 2\} \end{matrix}
1.3  23 − 1  \begin{matrix}\{1 − 23, 21 − 3, 2 − 31, 1 − 32\} or \\ \{1 − 23, 21 − 3, 2 − 31, 13 − 2\} or \\ \{1 − 23, 21 − 3, 2 − 31, 3 − 12\} or \\ \{1 − 23, 21 − 3, 2 − 31, 31 − 2\} \end{matrix}
1.3  23 − 1  \begin{matrix}\{1 − 23, 2 − 13, 2 − 31, 1 − 32\} or \\ \{1 − 23, 2 − 13, 2 − 31, 13 − 2\} or \\ \{1 − 23, 2 − 13, 2 − 31, 3 − 12\} or \\ \{1 − 23, 2 − 13, 2 − 31, 31 − 2\} \end{matrix}
1.4  1 − 32  \begin{matrix}\{1 − 23, 2 − 13, 2 − 31, 13 − 2\} or \\ \{1 − 23, 2 − 13, 23 − 1, 13 − 2\} \end{matrix}
1.4  1 − 32  \begin{matrix}\{1 − 23, 2 − 13, 13 − 2, 3 − 12\} or \\ \{1 − 23, 2 − 13, 13 − 2, 31 − 2\} \end{matrix}
1.4  1 − 32  \begin{matrix}\{1 − 23, 21 − 3, 2 − 31, 13 − 2\} or \\ \{1 − 23, 21 − 3, 23 − 1, 13 − 2\} \end{matrix}
1.4  1 − 32  \begin{matrix}\{1 − 23, 21 − 3, 13 − 2, 3 − 12\} or \\ \{1 − 23, 21 − 3, 13 − 2, 31 − 2\} \end{matrix}
1.4  1 − 32  \begin{matrix}\{1 − 23, 2 − 31, 13 − 2, 3 − 12\} or \\ \{1 − 23, 2 − 31, 13 − 2, 31 − 2\} \end{matrix}
1.4  1 − 32  \begin{matrix}\{1 − 23, 23 − 1, 13 − 2, 3 − 12\} or \\ \{1 − 23, 23 − 1, 13 − 2, 31 − 2\} \end{matrix}
1.5  12 − 3  \begin{matrix}\{1 − 23, 2 − 13, 2 − 31, 1 − 32\} or \\ \{1 − 23, 2 − 13, 2 − 31, 13 − 2\} or \\ \{1 − 23, 2 − 13, 23 − 1, 1 − 32\} or \\ \{1 − 23, 2 − 13, 23 − 1, 13 − 2\} \end{matrix}
1.5  12 − 3  \begin{matrix}\{1 − 23, 2 − 13, 2 − 31, 3 − 12\} or \\ \{1 − 23, 2 − 13, 2 − 31, 31 − 2\} or \\ \{1 − 23, 2 − 13, 23 − 1, 3 − 12\} or \\ \{1 − 23, 2 − 13, 23 − 1, 31 − 2\} \end{matrix}
1.5  12 − 3  \begin{matrix}\{1 − 23, 2 − 13, 1 − 32, 3 − 12\} or \\ \{1 − 23, 2 − 13, 1 − 32, 31 − 2\} \end{matrix}
1.6  12 − 3  \begin{matrix}\{1 − 23, 21 − 3, 23 − 1, 3 − 12\} or \\ \{1 − 23, 21 − 3, 23 − 1, 31 − 2\} \end{matrix}
1.6  12 − 3  \begin{matrix}\{1 − 23, 21 − 3, 2 − 31, 1 − 32\} or \\ \{1 − 23, 21 − 3, 23 − 1, 1 − 32\} \end{matrix}
1.6  12 − 3  \begin{matrix}\{1 − 23, 21 − 3, 2 − 31, 3 − 12\} or \\ \{1 − 23, 21 − 3, 2 − 31, 31 − 2\} \end{matrix}

Table 11: permutations avoiding five patterns

Enumerating sequence: {n}_{n≥1}

thanks to Proposition add the pattern to the representative
1.2  3 − 12  d2
1.3  23 − 1  d9
1.2  3 − 12  d6
1.2  3 − 12  d25
1.1  21 − 3  d14
1.1  21 − 3  d17
1.4  1 − 32  d16
1.1  21 − 3  d20
1.3  23 − 1  d18
1.4  1 − 32  d21
1.5  12 − 3  d9
1.5  12 − 3  d11
1.5  12 − 3  d1
1.5  12 − 3  d2
1.5  12 − 3  d10
1.5  12 − 3  d3
1.6  12 − 3  d12
1.6  12 − 3  d4
1.7  12 − 3  d28
1.7  12 − 3  d25

Table 12: permutations avoiding five patterns

Enumerating sequence: {F_{n}}_{n≥1}
thanks to Proposition add the pattern to the representative
1.5  12 − 3  e1
1.5  12 − 3  e3
1.1  21 − 3  e4
1.4  1 − 32  e10

Table 13: permutations avoiding five patterns

5 Conclusion: the cases of more than five patterns

Looking at the table of the conjectures by Claesson and Mansour [5], one can see that the case of six patterns is the unique, among the remaining ones, which presents some enumerating sequences not ultimately constant. More precisely, in this case there are three classes enumerated by the sequence {n}_{n≥1} and one class enumerated by {F_{n}}_{n≥1}. In order to find a representative for each of these classes, it is possible to consider the classes of permutations avoiding five patterns enumerated by the same sequence and apply one of the propositions in the Introduction. As far as the class enumerated by {F_{n}}_{n≥1} is concerned, the representative set U = {1 − 23, 2 − 13, 13 − 2, 21 − 3, 12 − 3, 1 − 32} is obtained by considering the patterns arising from the second line of Table 13 and applying Proposition 1.4. Note that the same above representative set can be obtained by considering the patterns of the third line of Table 13 and applying Proposition 1.6.

Starting from the set V = {1 − 23, 2 − 13, 1 − 32, 21 − 3, 12 − 3}, which are the forbidden patterns specified in the first line of Table 13, it is not possible to apply anyone of the propositions in the Introduction. Nevertheless, if π ∈ S(V), then in particular π avoids the patterns 1 − 23 and 1 − 32. Next, it is simple to see that such a permutation π avoids also 13 − 2, therefore π ∈ S_{n}(U) and |S_{n}(U)| = F_{n}.

The propositions in the Introduction are not even useful when the set of patterns Z = {1− 23, 1−32, 3−12, 31−2, 13−2}, arising from the fourth line of Table 13, is considered. In this case, if a pattern x ∈ M\Z is added to Z, then it can be proved that either |S_{n}(Z, x)| = n, or |S_{n}(Z, x)| = 0 (n ≥ 5), or |S_{n}(Z, x)| = n (n ≥ 2). A possible way to menage these proofs (one for each x ∈ M\Z) uses the ECO method, as Bernini, Ferrari and Pinzani made [3]. This is what we did but we do not show the details of the permutations’ construction, which can be easily recovered by the reader.

In order to prove the truth of the conjecture relating to the sequence {F_{n}}_{n≥1}, we observe that if a_{n} and b_{n} are the sequences enumerating S_{n}(A) and S_{n}(B), respectively, with A ⊆ B ⊆ M, then a_{n} ≥ b_{n}. Therefore, using the strategy mentioned in Subsection 1.2, the patterns specified in U can be obtained only from the forbidden patterns arising from Table 13.

A similar discussion should be carried out for the permutations avoiding six patterns enumerated by the sequence {n}_{n≥1}, starting from the patterns of Table 12. We omit it since it can simply be done by following the line of the case of the sequence {F_{n}}_{n≥1}. The complete solution of the conjectures of Claesson and Mansour requires that the same argument has to be applied for the ultimately constant sequences {0} and {1}, enumerating permutations avoiding six or more patterns. It should not be difficult to write down the tables summarizing the results relating to the permutations avoiding six patterns and use them, according to the usual strategy, for the results of the case of seven forbidden patterns. Then, the tables relating to the case of seven patterns should be used for the case of eight patterns, and so on.

Actually, we note that if the tables of the case of six patterns are not at our disposal it could be quite long to solve the case of seven forbidden patterns using the same strategy. Nevertheless, it does not seem necessary to provide the tables for all the remaining cases, since it is easy, given a set D ⊆ M of six or more forbidden patterns, to find the enumerating sequence of S(D). This sequence should be ultimately constant according to the conjectures by Claesson and Mansour [5] (aside from the already examined sequences {F_{n}}_{n≥1} and {n}_{n≥1} for the case of six forbidden patterns). Therefore, by means of the ECO method, for instance, it is straightforward to generate all the permutations of S(D), following the line which Bernini, Ferrari and Pinzani used in their paper [3], and find the relating sequence. Unless the conjectures of Claesson and Mansour are wrong, this kind of approach is quite fast every time |S(D)| is required.

We conclude with a hint for a possible further work. Mansour and Vainshtein [7] showed that the generating function for permutations avoiding 132 and any other classical pattern is rational. It should be interesting to generalize this result to some set of generalized patterns (the results of Claesson [4] stating that S(132) = S(13 − 2) or S(213) = S(2 − 13) could be used for a first effort in this direction). With a similar approach it should be possible to handle this kind of analysis mechanically.

Acknowledgments

A part of this paper has been written during a visit to the Centre de Recerca Matemática (CRM) in Barcelona. The hospitality and support of this institution is gratefully acknowledged.

References

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[2] E. Babson, E. Steingrímsson, Generalized permutation patterns and a classification of the Mahonian statistics, Sém. Lothar. Combin. 44 (2000), B44b.

[3] A. Bernini, L. Ferrari, R. Pinzani, Enumerating permutations avoiding three Babson-Steingrímsson patterns, Ann. Comb 9 (2005), 137–162.

[4] A. Claesson, Generalized pattern avoidance, European J. Combin. 22 (2001), 961–971.

[5] A. Claesson, T. Mansour, Enumerating permutations avoiding a pair of Babson-SteingrÝmsson patterns, Ars Combin. 77 (2005), 17–31.

[6] S. Elizalde, M. Noy, Consecutive patterns in permutations, Adv. in Appl. Math. 30 (2003), 110–125.

[7] T. Mansour, A. Vainshtein, Restricted 132-avoiding permutations, Adv, in Appl. Math 26 (2001), 258–269.

[8] S. Kitaev, Generalized patterns in words and permutations, Ph. D. Thesis, Chalmers University of Technology and Göteborg Univerity (2003).

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2000 Mathematics Subject Classification: Primary 05A15; Secondary 05A05. Keywords: generalized pattern avoidance, Babson-Steingrímsson patterns, permutations.

(Concerned with sequences \underline{A000027}, \underline{A000045}, \underline{A000079}, \underline{A000124}, \underline{A001405}, and \underline{A094373}.) Received April 27 2007; revised version received June 11 2007. Published in Journal of Integer Sequences, June 11 2007. Return to Journal of Integer Sequences home page.