#A2 INTEGERS 18 (2018) ON PATTERN AVOIDING INDECOMPOSABLE PERMUTATIONS

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1 #A INTEGERS 8 (08) ON PATTERN AVOIDING INDECOMPOSABLE PERMUTATIONS Alice L.L. Gao Department of Applied Mathematics, Northwestern Polytechnical University, Xi an, Shaani, P.R. China llgao@nwpu.edu.cn Sergey Kitaev Department of Computer and Information Sciences, University of Strathclyde, Glasgow, United Kingdom sergey.kitaev@cis.strath.ac.uk Philip B. Zhang College of Mathematical Science, Tianjin Normal University, Tianjin, P. R. China zhangbiaonk@63.com Received: 5//6, Revised: 8/3/7, Accepted: /6/7, Published: /6/8 Abstract Comtet introduced the notion of indecomposable permutations in 97. A permutation is indecomposable if and only if it has no proper prefi which is itself a permutation. Indecomposable permutations were studied in the literature in various contets. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of length 3. Further, we provide a recursive formula for enumerating... k-avoiding indecomposable permutations for k 3. Several of our results involve the descent statistic. We also provide a bijective proof of a fact relevant to our studies.. Introduction Let [n] = {,..., n} and S n be the set of permutations of [n]. Given = n S n, let i denote the smallest inde such that i is a permutation of [i ]. If i = n then is indecomposable; otherwise, is decomposable. For eample, 354 is indecomposable, while 354 is decomposable. Supported by the Fundamental Research Funds for the Central Universities(30070QD0). Supported by the National Science Foundation of China (Nos. 667, 7044).

2 INTEGERS: 8 (08) For a permutation of a set {a,..., a n }, the reduced form of, denoted red( ), is the permutation of [n] obtained from by replacing the i-th smallest element with i. For eample, red(537) = 34. For any permutation, we have = () () (k) for some k, where (i) is a permutation such that red( (i) ) is indecomposable for all apple i apple k. We say that a (i) is a component of. For eample, the permutation 3465 has components 3, 4 and 65. A (permutation) pattern is a permutation = k. We say that a permutation = n contains an occurrence of if there are apple i < < i k apple n such that i ik is order-isomorphic to, that is, if the reduced form of i ik is. If does not contain an occurrence of, we say that avoids. These patterns are referred to as classical patterns. For instance, the permutation 3567 contains several occurrences of the pattern 3, for eample, the subsequences 356 and 57, while this permutation avoids the pattern 3. A comprehensive introduction to the theory of patterns in permutations can be found in [9]. Other patterns of interest to us are vincular patterns, also known as generalized patterns [, 3], where some of the elements may be required to be adjacent in the permutation. We underline elements of a given pattern to indicate the elements that must be adjacent in any occurrence of the pattern. For eample, the permutation = 3654 contains four occurrences of the pattern 3, namely, the subsequences 6, 54, 354 and 54 (in each of these occurrences, the elements in corresponding to and 3 in the pattern stay net to each other). On the other hand, contains only one occurrence of the pattern 3, namely, 54. If all elements in an occurrence of a pattern are required to stay net to each other, which is indicated by underlying all elements in the pattern, such a pattern is called a consecutive pattern [5]. A classical statistic descent is just an occurrence of the pattern. Vincular patterns play an important role in the theory of patterns in permutations and words (see Sections 3.3 and 3.4 in [9] for details). The notion of indecomposable permutations (also known as irreducible permutations or connected permutations) was introduced by Comtet [9, 0]. Comtet was the first one to show the ordinary generating function for the number I n of indecomposable permutations of length n is X I n n = n= Pk 0 k!k. These numbers begin with,, 3, 3, 7, 46, 3447, 9093,... for n, and appear as the sequence A00339 in the On-Line Encyclopedia of Integer Sequences (OEIS) []. Indecomposable permutations appear in various contets in the literature, for eample, see [7, 8,, 3, 8]. In particular, in [7, Section 4], indecomposable permutations are used to define a bijection between 3- and 3-avoiding permutations (finding various bijections essentially between these sets was the subject of

3 INTEGERS: 8 (08) 3 several papers in the literature). Also, indecomposable pattern avoiding permutations are a key object in [8] to find a bijection between permutations in question and (, 0)-trees. Corollaries of this result include a number of equidistribution results on permutations, (, 0)-trees and certain types of planar maps. Finally, we note that indecomposable pattern avoiding permutations were first studied by Bóna in [3], where essentially 43-avoiding indecomposable permutations are enumerated (indecomposable permutations are defined up to reverse in [3]) and linked in a bijective way to (0, )-trees. In this paper, we study interrelations (taking into account the descent statistic, which comes for free ) between pattern avoiding permutations and their indecomposable counterparts for classical patterns of length 3 and 4, vincular nonconsecutive patterns of length 3, and the increasing classical pattern of arbitrary length (patterns of length are trivial). We use our results and known enumeration formulas for pattern avoiding permutations to enumerate indecomposable pattern avoiding permutations. Some of the obtained numbers appear in the OEIS, suggesting a number of bijective questions. The paper is organized as follows. In Section, we introduce generating functions to be studied in this paper and state known pattern avoiding results to be used. In Section 3, we study indecomposable permutations avoiding classical patterns of length 3 and 4, as well as the classical pattern... k for k 3. In Section 4, we study indecomposable permutations avoiding a vincular non-consecutive pattern of length 3, in particular, presenting a bijective result in Theorem 4.4. Finally, in Section 5, we summarize our enumerative results (see Table ) and discuss directions of further research.. Preliminaries Let A n and I n be the number of -avoiding permutations of [n] and -avoiding indecomposable permutations of [n], respectively. For 0 apple i apple n, let A n,i and I n,i be the number of -avoiding permutations of [n] and -avoiding indecomposable permutations of [n] with i descents, respectively. Thus, for n, we have A n = A n,i and I n = I n,i. i=0 Let A (), A (, q), I () and I (, q) be the generating functions for A n, A n,i, I n and I n,i, respectively. That is, n=0 i=0 i=0 X X A (, q) = A n,i n q i, I (, q) = I n,i n q i, n= i=0

4 INTEGERS: 8 (08) 4 X X A () = A (, ) = A n n and I () = I (, ) = I n n. n=0 If for patterns and, An = An for all n 0 then and are Wilf-equivalent. For a permutation = n, its reverse is the permutation r( ) = n n and its complement is the permutation c( ) = (n+ )(n+ ) (n+ n ). For eample, if = 345 then r( ) = 543 and c( ) = 345. The reverse and complement operations are called trivial bijections. It is easy to see that for any pattern, this pattern is Wilf-equivalent to r( ) and c( ). Another useful property of trivial bijections is that their composition preserves the property of being decomposable (and thus the property of being indecomposable), which is easy to see. One of concerns in this paper is to find interrelations between I (, q) and A (, q) for certain s. We note that throughout this paper we implicitly use the fact that an occurrence of a descent cannot start in one component of a permutation and end in another one. In the rest of this section we review a number of permutation pattern avoidance results relevant to this paper. Lemma. ([0]). For S 3 and n 0, A n = C n = n+ number. Thus, A () = C() = p 4, n n n=, the n-th Catalan the generating function for the Catalan numbers satisfying C() C() + = 0. For n 0, the Catalan numbers begin with,,, 5, 4, 4,..., which is the sequence A00008 in the OEIS []. The net result links the well-known Bell numbers to pattern avoiding permutations. The Bell numbers begin with,,, 5, 5, 5, 03,... for n 0, and this is the sequence A0000 in the OEIS []. Lemma. ([6]). When {3, 3, 3, 3}, for n 0, we have A n = B n, where B n is the n-th Bell number, which is the number of set partitions of [n]. When {3, 3}, for n 0, we have A n = C n. We net turn our attention to classical patterns of length 4. Table presents three Wilf-equivalence classes in this case. A 34 n for n 0 begins with,,, 6, 3, 03,... (this is sequence [, A00580]), and we have the following lemma.

5 INTEGERS: 8 (08) 5 34,43, 43,34,34,43, 43,34,34,43, 43,34 34,43,34,43, 43,34,34,43, 43, ,43 Table : The three Wilf-equivalence classes for pattern avoidance of length 4. Spaces on a line are used to group patterns equivalent via trivial bijections. Lemma.3 ([6]). For {34, 43, 43, 34, 34, 43, 43, 34, 34, 43, 43, 34}, we have E() := A () = + 5 ( 9) ( ) 4 F 4, 3 4 ; ; 64. ( )( 9) 3 Moreover, for n, we have E n := A n = k=0 k k n k 3k + k kn n + (k + ) (k + )(n k + ). An eact enumeration for 34-avoiding permutations and the corresponding generating function are given by Bóna [3]. The corresponding sequence for n 0 begins with,,, 6, 3, 03, 5,... (A0558 in []) and the following lemma holds. Lemma.4 ([3]). For {34, 43, 34, 43, 43, 34, 34, 43, 43, 34}, we have F () := A () = Moreover, for n ( 8)3/ = ( 8) 3/ ( + ) 3 + ( + ) 3., we have F n := A n = 7n 3n ( ) n + 3 ( ) n i i+ (i 4)! n i +. i!(i )! However, no formula for A 34 n is known, only a recurrence relation is discovered [], and an algorithm for counting the number of 34-avoiding permutations was given in [, 7]. For recent developments on the bounds, see [, 4]. The corresponding sequence for n 0 begins with,,, 6, 3, 03, 53, 76,...; see A0655 in []. i= 3. Indecomposable Permutations Avoiding Classical Patterns Patterns are permutations, and we distinguish two cases according to whether or not they are decomposable. We start with an easier case.

6 INTEGERS: 8 (08) Indecomposable Patterns Here we deal with the following patterns: 3, 3, 3, 34, 43, 43, 34, 34, 34, 34, 43, 43, 43, 43, 43, 43. We first establish a property holding for any indecomposable pattern. Lemma 3.. If is an indecomposable pattern, then I (, q) satisfies I (, q) = A (, q). Proof. For any permutation, an occurrence of cannot start in one component and end in another one, which would contradict being irreducible. Similarly, a descent cannot start in one component and end in another one. Hence, the generating function for -avoiding permutations with k components is [I (, q)] k and X A (, q) = + (I (, q)) k = I (, q), k= where + corresponds to the empty permutation. This gives the desired result. Combining Lemma 3. (q = ) and Lemma. we obtain the following theorem, which can also be derived, e.g. from considerations in [7]. Theorem 3.. For {3, 3, 3}, we have I () = p 4. () Thus, for n, I n = C n, the (n )-th Catalan number. Combining Lemma 3. (q = ) and Lemma.4 we obtain the following theorem established in [3]. Theorem 3.3. For {43, 43, 34, 43, 43, 34}, we have I () = ( 8) 3/. 3 The initial values for I n in this case are,, 3,, 56, 88, 584, 95,... for n, and this is the sequence A00057 in the OEIS []. Similarly, one can combine Lemma 3. (q = ) and Lemma.3 to obtain a formula for I (), where {43, 34, 43, 34, 43, 34}. The initial values for I n in this case are,, 3,, 56, 89, 603, 939,... for n, and this sequence is not in the OEIS [].

7 INTEGERS: 8 (08) 7 However, we cannot obtain a formula for I 43 () using Lemma 3. because no formula is known for A 43 (). The initial values for In 43 are,, 3,, 56, 89, 604, 945,... for n, and this sequence is not in the OEIS []. 3.. Decomposable Patterns The patterns here we deal with are 3, 3, 3 and 34, 43, 34, 34, 43, 43, 34, 43, 34, 34, Decomposable Patterns of Length 3 Pattern 3. We first give a description of 3-avoiding decomposable permutations. Lemma 3.4. Let = () 0 S n, where () is the component in formed by the elements in {,..., i }. Then is a 3-avoiding decomposable permutation if and only if () = i (i ), 0 = n(n ) (i + ) and apple i apple n. Proof. The backward direction is straightforward to see since no occurrence of the pattern 3 can start in (). For the forward direction, since = n is decomposable, one must have apple i apple n. If i < j for some apple i < j apple i then i j n is an occurrence of the pattern 3; contradiction. If i < j for some i + apple i < j apple n then i j is an occurrence of the pattern 3; contradiction. Thus, we obtain the desired result. Net we derive a relation between I 3 (, q) and A 3 (, q), which will give formulas for I 3 () and In 3. The initial values for In 3 for n begin with,, 3,, 38, 7, 43,.... This sequence does not appear in the OEIS []. Theorem 3.5. We have that A 3 (, q) = I 3 (, q) + +, () ( q) and for n, I 3 () = p 4 ( ), (3) I 3 n = C n (n ). (4)

8 INTEGERS: 8 (08) 8 Proof. By Lemma 3.4, the generating function for all 3-avoiding decomposable permutations is P i i q i P j j q j = ( q). Since any 3-avoiding permutation is either indecomposable, or decomposable, or the empty permutation, we obtain the relation given by (). Letting q = in () and using Lemma., we obtain (3). Finally, by Lemma. and the fact that ( ) = P n (n )n, we obtain (4). This completes the proof. We note that (4) appears in Proposition 9 in [4]. Patterns 3 and 3. We begin with a description of 3-avoiding decomposable permutations. Lemma 3.6. Let = () 0 S n, where () is the component in formed by the elements in {,..., i }. Then is a 3-avoiding decomposable permutation if and only if () is a 3-avoiding decomposable permutation, apple i apple n, and 0 = (i + )(i + ) n. Proof. The backward direction is easy to see since an occurrence of the pattern 3 cannot start in () in this case. For the forward direction, since is decomposable, we have apple i apple n. Moreover, since is 3-avoiding, () must be 3-avoiding. Finally, if i > j for i + apple i < j apple n then i j is an occurrence of the pattern 3, which is a contradiction. This completes the proof. Net we find a relation between A (, q) and I (, q) for {3, 3}, which will give us formulas for I () and I n. The initial values for In 3 = In 3 for n begin with,, 3, 9, 8, 90, 97, 00,..., which is the sequence A00045 in []. Theorem 3.7. For I =, and for n, {3, 3}, we have that A (, q) = + I (, q), (5) p 4 I () = ( ), (6) I n = C n C n. (7) Proof. Let = 3. By Lemma 3.6, the generating function for all 3-avoiding decomposable permutations is I 3 (, q) P j j = I3 (, q). Similarly to the proof of Theorem 3.5, we have A 3 (, q) = + I 3 () + I3 (, q) = + I3 (, q)

9 INTEGERS: 8 (08) 9 giving (5) for = 3. Letting q = in (5) and using Lemma., we obtain I 3 () = ( )(A 3 () ) p 4 = ( ) giving (6) for = 3. From (6), I 3 = and, for n (7) follows for = 3. Finally, since the composition of reverse and complement preserves the property of being irreducible, and this composition applied to 3 gives 3, we have that (5), (6) and (7) hold for = 3. This completes the proof. Note that (7) follows directly from Lemma 3.6. Indeed, in a decomposable 3- avoiding permutation, the largest element n must be the rightmost element, and the number of such permutations is C n, while the number of all 3-avoiding permutations of length n is C n. Also, note that (7) appears in Proposition 9 in [4] Decomposable Patterns of Length 4 Recall that when applying the composition of reverse and complement, the property of being decomposable is retained. Applying that composition to 43, 34 and 34 yields the original pattern. For the remaining eight decomposable patterns this operation gives I 34 (, q) = I 43 (, q), I 34 (, q) = I 34 (, q), I 34 (, q) = I 43 (, q), and I 34 (, q) = I 43 (, q). Thus, we only need to consider seven decomposable patterns of length 4. Patterns 34 and 34. We start with a description of 34-avoiding and 34- avoiding decomposable permutations. Lemma 3.8. Let = () 0 S n, where () is a permutation of {,..., i }. Then is a 34-avoiding (resp., 34-avoiding) decomposable permutation if and only if () is 3-avoiding (resp., 3-avoiding) and 0 is 34-avoiding (resp., 34-avoiding). Proof. For the backward direction, because () is 3-avoiding (resp., 3-avoiding), at most two elements in a possible occurrence of the pattern 34 (resp., 34) can be in (). But then 0 contains an element smaller than an element in (), which is impossible, and thus is 34-avoiding (resp., 34-avoiding). For the forward direction, since = n is decomposable, we have apple i apple n. Also, clearly 0 is 34-avoiding (resp., 34-avoiding). Now, if () would contain an occurrence of the pattern 3 (resp., 3) then together with n it would form an occurrence of the pattern 34 (resp., 34); contradiction. Thus () is 3-avoiding (resp., 3-avoiding). This completes the proof.

10 INTEGERS: 8 (08) 0 The sequence for I 34 n and I 34 n for n begin with,, 3, 3, 65, 350, 979, 6,..., which does not appear in the OEIS []. Theorem 3.9. We have and A 34 (, q) = + I 34 (, q) + I 3 (, q) A 34 (, q) (8) A 34 (, q) = + I 34 (, q) + I 3 (, q) A 34 (, q). (9) Further, for {34, 34}, we have that I () = p 3 4 +, ( 8) 3/ with I =, and for n, where F n is defined in Lemma.4. I n = F n C i F n i, i=0 Proof. By Lemma 3.8, the generating function for 34-avoiding decomposable permutations is I 3 (, q)(a 34 (, q) ), where corresponds to ecluding the empty permutation as a possibility for 0. Similarly, the generating function for 34-avoiding decomposable permutations is I 3 (, q)(a 34 (, q) ). Note that each -avoiding permutation is either the empty permutation, or an indecomposable permutation or a decomposable one. This observation shows (8) and (9). Let q = in (8). Combing this with Lemma.4 and (), we obtain I 34 () = I 3 () A 34 () (0) = ( C()) (F () ) () = p () ( 8) 3/ Hence, by (), we have that I 34 = and for n, I 34 n = F n + C n C i F n i i=0 = F n C i F n i. i=0 It is straightforward to provide essentially the same derivations for the case of 34- avoiding indecomposable permutations, which completes the proof.

11 INTEGERS: 8 (08) Pattern 34. We begin with a description of 34-avoiding decomposable permutations. Our proof of the net lemma is similar to the proof of Lemma 3.8 and thus is omitted. Lemma 3.0. Let = () 0 S n, where () is a permutation of {,..., i }. Then is a 34-avoiding decomposable permutation if and only if () is 3- avoiding and 0 is a 34-avoiding. The initial values I 34 n for n begin,, 3, 3, 65, 35, 999, 87,... and this sequence is not in the OEIS []. Theorem 3.. We have A 34 (, q) = + I 34 (, q) + I 3 (, q) A 34 (, q). (3) Moreover, for E() and E n defined in Lemma.3, we have I 34 () = p 4 + (E() ), I 34 = and for n I 34 n = E n C i E n i. i=0 Proof. We can proceed similarly to the proof of Theorem 3.9 to prove (3). Further, assuming that q = in (3), one can apply Lemma.3 and (), to obtain I 34 () = I 3 () A 34 () = ( C()) (E() ) = p 4 + (E() ). From the last derivation, the formula for I 34 n holds. Pattern 43. We begin with a description of 43-avoiding decomposable permutations. Lemma 3.. Let = () 0 S n, where () is a permutation of {,..., i }. Then is a 43-avoiding decomposable permutation if and only if one of the following two conditions holds:

12 INTEGERS: 8 (08) () = and 0 is 43-avoiding. apple i apple n, () is 43-avoiding and 0 = (i + )(i + ) n. Proof. For the forward direction, since is decomposable, we have apple i apple n. There are two cases to consider: i =. It is clear that () = in this case does not a ect 0, so 0 must be a 43-avoiding permutation of {,..., n}. apple i apple n. Since () is indecomposable of length at least, there eist apple j < j apple i such that j > j. But then to avoid an occurrence of the pattern 43 involving j and j, 0 must be the increasing permutation (i + )(i + ) n. The backward direction is easy to see using similar considerations as above. The initial values I 43 n for n are,, 3, 3, 63, 330, 838, 0758,..., and this sequence is not in the OEIS []. Theorem 3.3. We have A 43 (, q) = + I 43 (, q) + A 43 (, q) + I43 (, q). (4) Moreover, for E() and E n defined in Lemma.3, we have I 43 n = and for n, I 43 () = ( ) E() +, I 43 n = E n E n + E n. Proof. By Lemma 3., (4) follows. Lemma.3, it follows that Further, letting q = in (4) and using from which the formula for I 43 n I 43 () = ( ) A 43 () + = ( ) E() +, follows. Pattern 34. We first give a description of 34-avoiding decomposable permutations. Lemma 3.4. Let = () 0 S n, where () is a permutation of {,..., i }. Then is a 34-avoiding decomposable permutation if and only if one of the following two conditions holds:

13 INTEGERS: 8 (08) 3 () = and 0 is 34-avoiding. apple i apple n, () is 3-avoiding, and 0 = n(n ) (i + ). Proof. For the forward direction, since = n apple i apple n. There are two cases to consider: is decomposable, we have i =. It is clear that () does not a ect the rest of the permutation, and 0 must be a 34-avoiding permutation of {,..., n}. apple i apple n. Then () must be a 3-avoiding permutation, or else it would form an occurrence of the pattern 34 with n. Moreover, since () is indecomposable, there eist apple j < j apple i such that j > j. But then to avoid an occurrence of the pattern 34 involving j and j, 0 must be the decreasing permutation n(n ) (i + ). The backward direction is not di cult to see using the considerations above. Initial values for I 34 n are,, 3, 3, 67, 369, 7, 578,... for n, and this sequence is not in the OEIS []. Theorem 3.5. We have A 34 (, q) = + I 34 (, q) + A 34 (, q) + q I3 (, q). (5) Moreover, for E() and E n defined in Lemma.3, we have I 34 = and for n, I 34 () = ( )E() C() + +, I 34 n = E n E n C n +. Proof. The identity (5) follows from Lemma 3.4. Further, setting q = in (5), and applying Theorem 3.7 and Lemma.3, one has I 34 () = ( )A 34 () ( ) I3 () + = ( )A 34 () ( ) (C() ) + = ( )E() C() + + From this, we have the desired formula for I 34 n..

14 INTEGERS: 8 (08) 4 Pattern 34. We first give a description of 34-avoiding decomposable permutations. Lemma 3.6. Let = () 0 S n, where () is a permutation of {,..., i }. Then is a 34-avoiding decomposable permutation if and only if () is 3- avoiding and 0 is 3-avoiding. Proof. For the forward direction, since = n is decomposable, we have apple i apple n. Moreover, () is 3-avoiding, or else, along with n an occurrence of the pattern 34 would be formed. Also, 0 is 3-avoiding or else, along with an occurrence of the pattern 34 would be formed. The backward direction is not di cult to see. Initial values for In 34 not in the OEIS []. Theorem 3.7. We have Also, are,, 3, 3, 69, 396, 355, 4363,..., and this sequence is A 34 (, q) = + I 34 (, q) + I 3 (, q) A 3 (, q). (6) Moreover, I 34 = and for n, I 34 () = A 34 () ( ) (C() ). I 34 n = A 34 n C n+ + 3C n C n. Proof. The identity (6) follows from Lemma 3.6. Further, setting q = in (6) and applying Lemma. and Theorem 3.7, we obtain I 34 () = A 34 () I 3 () A 3 () = A 34 () ( ) (C() ) apple C() = A 34 () ( ) C() +. Hence, I 34 = and for n, This completes the proof. I 34 n = A 34 n (C n+ C n ) + (C n C n ) = A 34 n C n+ + 3C n C n.

15 INTEGERS: 8 (08) 5 Pattern 34. follows. Decomposable 34-avoiding permutations can be described as Lemma 3.8. Let = () 0 S n, where () is a permutation of {,..., i }. Then is a 34-avoiding decomposable permutation if and only if one of the following two conditions holds: () = i (i ), 0 is 3-avoiding and apple i apple n, or () 6= i (i ) is 3-avoiding, 0 = n(n ) (i + ), and 3 apple i apple n. Proof. Since is 34-avoiding, then () must be 3-avoiding, or else there would be an occurrence of the pattern 34 involving an element in 0. Thus, the longest increasing sequence in () is at most of length. There are two cases to consider. () = i (i ). Then, clearly, 0 must be 3-avoiding. The longest increasing subsequence in () is eactly of length. But then, since () is indecomposable, we have i > and 0 must be -avoiding, that is, 0 = n(n ) (i + ). This completes the proof. Initial values for I 34 n are,, 3, 3, 69, 400, 390, 4545,... for n, and this sequence is not in the OEIS []. Theorem 3.9. We have A 34 (, q) = + I 34 (, q) + q A3 (, q) + q I 3 (, q). (7) q Also, for E() defined in Lemma.3, I 34 () = E() C() + 3 ( ) 3 + ( ) +. Moreover, I 34 = and for n and for E n defined in Lemma.3, n = E n C i + n n 4. I 34 i=0

16 INTEGERS: 8 (08) 6 Proof. The identity (7) follows from Lemma 3.8. Further, setting q = in (7), and applying Lemmas. and.3 and Theorem 3.5, we have I 34 () = A 34 () A3 () I 3 () = E() = E() C() ( ) C() + 3 ( ) 3 + ( ) + Hence, it follows that I 34 = and for n n = E n C i + n n + 4 I 34 This completes the proof. i=0 = E n C i Pattern... k with k 3 i= n(n ). Here we consider patterns of the form... k, where k 3, which generalizes our considerations for patterns 3 and 34. First, we give a description of... k- avoiding decomposable permutations in the following lemma, whose proof is trivial and thus is omitted. Lemma 3.0. If = () 0 is a... k-avoiding decomposable permutation of length n, where () is a permutation of {,..., i }. Then there eists m, apple m apple k, such that the longest increasing subsequence in () is eactly of length m and 0 is... (k m)-avoiding. Now we can enumerate... k-avoiding indecomposable permutations. Corollary 3.. We have I...k (, q) =A...k (, q) kx m= Proof. By Lemma 3.0, we have A...k (, q) = + I...k (, q) + kx m= from which the result follows. I...(m+) (, q) I...m (, q) A...(k m) (, q) I...(m+) (, q) I...m (, q) A...(k m) (, q),..

17 INTEGERS: 8 (08) 7 For eample, when k = 3, we have and hence A 3 () = + I 3 () + I () I () A () = + I 3 () +, I 3 () = A 3 () ( ) = p 4 ( ), which coincides with Theorem 3.5. Note that we used the facts that I () () = 0 and A () = I () () = + + =. When k = 4, we have A 34 () = +I 34 ()+ I () I () A 3 () + I 3 () I () A (), and hence I 34 () = E() C() ( ), which coincides with Theorem 3.9. Note that we used the fact that A 3 () = C(). 4. Indecomposable Permutations Avoiding Vincular Non-consecutive Patterns of Length 3 For a pattern of the form abc, its reverse complement gives a pattern of the form yz. Thus, since the composition of reverse and complement preserves the property of being indecomposable, we only need to consider si cases of vincular patterns of length 3, which are 3, 3, 3, 3, 3, and 3. Two of these cases can be reduced to classical pattern-avoidance. Indeed, it was shown in [6] that a permutation avoids the pattern 3 if and only if it avoids the pattern 3. Applying the complement operation, this implies that a permutation avoids 3 if and only if it avoids 3. Thus, In 3 = In 3 and In 3 = In 3 and Theorems 3.7 and 3. can be applied, respectively. Pattern 3. We first give a description of 3-avoiding decomposable permutations. Lemma 4.. Let = () 0 S n, where () is a permutation of {,..., i }. Then = n is a 3-avoiding decomposable permutation if and only if () = i and 0 = n(n ) (i + ) for apple i apple n, where i is a 3-avoiding permutation of {, 3,..., i }.

18 INTEGERS: 8 (08) 8 Proof. Since is decomposable, we have apple i apple n. It is clear that () is a 3- avoiding indecomposable permutation. We claim that i =, since otherwise, i and i + will form the pattern 3. Further, clearly i + > i + > > n, or else there would be an occurrence of the pattern 3 involving. On the other hand, it is easy to see that if () and 0 satisfy the conditions then is 3-avoiding. This completes the proof. Initial values for I 3 n for n are,, 3,, 43, 79, 80,... and this sequence is not in the OEIS []. Theorem 4.. We have Also, A 3 (, q) = + I 3 (, q) + A 3 (, q) q +. (8) q I 3 () = B(), where B() is the generating function for the Bell numbers. Moreover, I 3 = and for n, I 3 n = B n B i. Proof. By Lemma 4., the generating function for 3-avoiding decomposable permutations is q A 3 (, q) q +, hence (8) follows. Letting q = in (8), we have I 3 () = A 3 (). Combining with Lemma., we obtain that i=0 I 3 () = B(). Together with the fact that = P i 0 i, we have I 3 = and for n, This completes the proof. I 3 n = B n B i. i=0 Pattern 3. We first give a description of 3-avoiding decomposable permutations.

19 INTEGERS: 8 (08) 9 Lemma 4.3. Let = () 0 S n, where () is a permutation of {,..., i }. Then = n is a 3-avoiding decomposable permutation if and only if () is 3-avoiding and 0 = i (i + ) n for apple i apple n. Proof. If is 3-avoiding, then clearly () and 0 are both 3-avoiding. Moreover, we must have i + < i + < < n, or else there would be an occurrence of the pattern 3 involving. On the other hand, it is clear that if () and 0 satisfy the given conditions then is 3-avoiding, which completes the proof. Initial values for I 3 n for n are,, 3, 0, 37, 5, 674,..., which are essentially the sequences A and A38378 in the OEIS [] that have several combinatorial interpretations. In particular, this sequence counts 3-avoiding permutations that end with a rise, that is, with an occurrence of the pattern, which leads us to the following theorem. Theorem 4.4. For n, the number of 3-avoiding indecomposable permutations in S n is equal to that of 3-avoiding permutations in S n that end with a rise. Proof. Let I and R be the first and the second sets, respectively, in the statement of the theorem. We provide a recursive bijection f from I to R proving the theorem with the base case f() =. The set of 3-avoiding permutations can be subdivided into three disjoint subsets: S, all 3-avoiding permutations ending with ; S, all 3-avoiding permutations ending with n; S 3, all other 3-avoiding permutations. It is straightforward to see that the elements to the right of in a 3-avoiding permutation must be in increasing order. But then in S 3, n must be to the left of. Thus, a permutation in S 3 belongs to both I and R and we map it to itself. Further, it is easy to see that S is a subset of I but it is disjoint from R, while S is a subset of R but it is disjoint from I. For a permutation = n S we define its image recursively as f( ) = f(( ) ( n ))n. The map f described by us is easy to see to be a bijection. This completes the proof. Net we enumerate 3-avoiding indecomposable permutations. Theorem 4.5. We have A 3 (, q) = + I 3 (, q) + I3 (, q). (9) Also, I 3 () = ( )B(), where B() is the generating function for the Bell numbers. Moreover, In 3 = and for n In 3 = B n B n.

20 INTEGERS: 8 (08) 0 Proof. By Lemma 4.3, the generating function for 3-avoiding decomposable permutations is I3 (, q) from which (9) follows. Letting q = in (9) and combining with Lemma., we obtain that Hence, it follows that for n I 3 () = ( )A 3 () = ( )B().. This completes the proof. I 3 n = B n B n Patterns 3 and 3. We first give a description of 3-avoiding and 3-avoiding decomposable permutations. Lemma 4.6. Let = () 0 S n, where () is a permutation of {,..., i }. Then is 3- (resp., 3-)avoiding if and only if () and 0 are both 3- (resp., 3-)avoiding. Initial values for I 3 n = I 3 n for n are,,, 6,, 9, 46,..., and this is the sequence A in the OEIS [] that has several combinatorial interpretations. In particular, this sequence counts the number of irreducible set partitions of [n], which can be easily seen from the bijections in [6]. For more information on irreducible set partitions, see [5]. Theorem 4.7. We have I 3 () = I 3 () = B(). Proof. Since 3 is an irreducible pattern, Lemma 3. can be applied to obtain A 3 () = I 3 (). The desired result now follows from Lemma.. 5. Concluding Remarks The notion of indecomposable permutations proved to be useful in various contets, e.g. in obtaining non-trivial enumeration and equidistribution results on permutations [8]. In this paper, we gave a compete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of length 3; see Table for a summary of these results. Also, we provided a recursive formula for enumerating... k- avoiding indecomposable permutations for k 3 (see Corollary 3.). The descent statistic is taken into account in several of our results.

21 INTEGERS: 8 (08) I () p 4 OEIS 3, 3, 3 A00008 p 3, 3 ( ) 4 A , 43, 34, 43, 43, 34 34, 34 p 34 p 4 ( ) ++8 +( 8) 3/ ( 8) 3/ A00057 p 4 + (E() ) 43 ( ) E() + 34 ( )E() C() A 34 () ( ) (C() ) 3 34 E() C() + ( ) + 3 ( ) + 3 B() 3 ( )B() 3, 3 B() A A38378 A Table : A summary of the avoidance results in this paper. The definitions of the functions E(), C(), B() and A 34 () can be found in Section and Theorem 4.. A natural direction of further research is in etending our studies of indecomposable permutations to other patterns, e.g. vincular patterns of length 4. Also, one can look at avoiding more than one pattern at the same time. Other statistics can be included in enumerative results. Finally, one can establish a number of bijective results linking pattern avoiding indecomposable permutations to other structures (Theorem 4.4 is one such eample). For instance, the sequence A in the OEIS [] has many interesting combinatorial interpretations that one could try to link in a bijective way to 3- avoiding indecomposable permutations. Acknowledgments. The second author is grateful to the administration of the Center for Combinatorics at Nankai University for their hospitality during the author s stay in November December 05. Also, the authors are grateful to Filippo

22 INTEGERS: 8 (08) Disanto for brining to our attention the paper [4], and to the anonymous referee for many useful suggestions on how to improve our paper. References [] E. Babson and E. Steingrímsson, Generalized permutation patterns and a classification of the Mahonian statistics, Sém. Lothar. Combin. 44.B44b(000), [] D. Bevan, Permutations avoiding 34 and patterns in lukasiewicz paths, J. Lond. Math. Soc. () 9(05), 05. [3] M. Bóna, Eact enumeration of 34-avoiding permutations: a close link with labelfed trees and planar maps, J. Combin. Theory Ser. A 80(997), [4] M. Bóna, A new record for 34-avoiding permutations, Eur. J. Math. (05), [5] W. Y. C. Chen, T. X. S. Li, and D. G. L. Wang, A bijection between atomic partitions and unsplitable partitions, Electron. J. Combin. 8(0),.7. [6] A. Claesson, Generalized pattern avoidance, European J. Combin. (00), [7] A. Claesson and S. Kitaev, Classification of bijections between 3- and 3-avoiding permutations, Séminaire Lotharingien de Combinatoire B60d(008), 30. [8] A. Claesson, S. Kitaev, and E. Steingrímsson, Decompositions and statistics for beta(,0)- trees and nonseparable permutations, Adv. Appl. Math. 4(009), [9] L. Comtet, Sur les coe cients de l inverse de la série formelle P n!t n, C. R. Acad. Sci. Paris Sér. A-B 75(97), A569 A57. [0] L. Comtet, Advanced combinatorics, D. Reidel Publishing Co., Dordrecht, enlarged edition, 974. [] A. R. Conway and A. J. Guttmann, On 34-avoiding permutations, Adv. Appl. Math. 64(05), [] R. Cori, Hypermaps and indecomposable permutations, European J. Combin. 30(009), [3] R. Cori, Indecomposable permutations, hypermaps and labeled Dyck paths, J. Combin. Theory Ser. A 6(009), [4] F. Disanto, Some statistics on the hypercubes of Catalan permutations, J. Integer Sequences 8(05), 5... [5] S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30(003), 0 5. [6] I. M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53(990), [7] F. Johansson and B. Nakamura, Using functional equations to enumerate 34-avoiding permutations, Adv. Appl. Math. 56(04), [8] A. King, Generating indecomposable permutations, Discrete Math. 306(006),

23 INTEGERS: 8 (08) 3 [9] S. Kitaev, Patterns in permutations and words, Monographs in Theoretical Computer Science, An EATCS Series, Springer, Heidelberg, 0. With a foreword by Je rey B. Remmel. [0] P. A. MacMahon, Combinatory analysis, Vol. I, II (bound in one volume), Dover Phoeni Editions, Dover, Mineola, 004. Reprint of An introduction to combinatory analysis (90) and Combinatory analysis. Vol. I, II (95, 96). [] D. Marinov and R. Radoičić, Counting 34-avoiding permutations, Electron. J. Combin. 9(00/03),.3. Permutation patterns (Otago, 003). [] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at [3] E. Steingrímsson, Generalized permutation patterns a short survey, In Permutation patterns, volume 376 of London Math. Soc. Lecture Note Ser., pages 37 5, Cambridge Univ. Press, Cambridge, 00.