Permutations avoiding an increasing number of length-increasing forbidden subsequences

A permutation (cid:0) is said to be ✁ –avoiding if it does not contain any subsequence having all the same pairwise comparisons as ✁ . This paper concerns the characterization and enumeration of permutations which avoid a set ✂☎✄ of subsequences increasing both in number and in length at the same time. Let ✂ ✄ be the set of subsequences of the form “ ✆✞✝✠✟☛✡✌☞✎✍✏✝✠✟✑✡✓✒✔✍ ”, ✆ being any permutation on ✕✖☞✘✗✚✙✛✙✛✙✛✗✜✟✣✢ . For ✟✥✤✦☞ the only subsequence in ✂★✧ is ☞✎✒✪✩ and the ☞✎✒✪✩ –avoiding permutations are enumerated by the Catalan numbers; for ✟✫✤✬✒ the subsequences in ✂☛✭ are ☞✚✒✔✩✪✮ , ✒✣☞✯✩✔✮ and the ( ☞✎✒✔✩✪✮✰✗✱✒✲☞✚✩✔✮✖✍ –avoiding permutations are enumerated by the Schr¨oder numbers; for each other value of ✟ greater than ✒ the subsequences in ✂ ✄ are ✟✰✳ and their length is ✝✠✟✴✡✵✒✘✍ ; the permutations avoiding these ✟✰✳ subsequences are enumerated by a number sequence ✕✎✶✰✷✸✢ ✷✣✹ ✧ such that ✺✻✷✽✼✾✶✿✷✫✼✾❀❁✳ , ✺✻✷ being the ❀ –th Catalan number. For each ✟ we determine the generating function of permutations avoiding the subsequences in ✂❂✄ , according to the length, to the number of left minima and of non-inversions.


Introduction
The study of permutations represents an interesting and relevant discipline in Mathematics which began with Euler who first analyzed permutation statistics related to the study of parameters different from their length [16].MacMahon in [24] further developed this vast field but meaningful progress has been only made in the last thirty years.
Recently, the new problems coming from Computer Science led to the development of the concept of permutations with forbidden subsequences.They arise in sorting problems [10,22,33,36,37], in the analysis of regularities in words [4,23], in particular instances of pattern matching algorithms optimization [8]; just to mention some examples.The enumeration of permutations with specific forbidden subsequences has also applications in areas such as Algebraic Geometry and Combinatorics.The ❃ ❅❄ ✪❆ ❈❇ - avoiding permutations, called vexillary permutations, are relevant to the theory of Schubert polynomials.In Combinatorics, permutations with forbidden subsequences play an important role as they present bijections with a great number of non-trivial combinatorial objects [13,14,15,18,20,21] and moreover their enumeration gives rise to classical number sequences.The -th Catalan number is the common value of permutations with a single forbidden subsequence of length three [22].More precisely Knuth shows that ❇ ❄ ✘❃ -avoiding permutations are the one stack sortable permutations.The problem of avoiding more than one restriction was first studied by Simion and Schmidt [34] and they determined the number of permutations avoiding two or three subsequences of length three.
As far as forbidden subsequences of length four are concerned, new enumeration results concern the subsequence ❄ ✘❃ ✿❇ ✣❆ [17], ❄ ✔❇ ✣❆ ✸❃ [6] and all the ones behaving identically [1,35,36], while for ❄ ✔❇ ❈❃ ✲❆ -avoiding permutations the only result is proved by Bóna in [5] and it gives only a numerical lower bound.Permutations avoiding some couples of subsequences of length four give the Schröder numbers [20]; results concerning permutations avoiding more than one forbidden subsequence of length four exist; we refer to [20] for an exhaustive survey of the results available on permutations with forbidden subsequences.
Regarding permutations avoiding a single subsequence of length greater than four the most important result solves the problem of one increasing subsequence of any length giving an asymptotic value of the number of permutations avoiding the subsequence ❄ ✂✁ ✄✁ ☎✁ ✆ ✞✝ ❄ [30].In [11] Chow and West study ✟ ❄ ✘❃ ✿❇ ✡✠ ✆ ✁ ☎✁ ✄✁ ✪❄ ✟ ✆ ☛✝ ❄ ✌☞ ✍☞ -avoiding permutations; their generating functions can be expressed as a quotient of modified Chebyshev polynomials and they give rise to number sequences lying between the well-known Fibonacci and Catalan numbers.In [3] the authors study permutations where the latter forbidden subsequence means that the subsequence Their generating functions are algebraic and give rise to sequences of numbers lying between the well-known Motzkin and Catalan numbers involving the left-Motzkin factor numbers as a particular case.
A natural generalization of sequence avoidance is the restricted sequence inclusion.In this case a prescribed number of occurences of a sequence in the permutations is required.Noonan [28] and Bona [7] determined a simple expression for the number of permutations containing exactly one 123, respectively one 132 sequence.Robertson [31] proved that the number of ❄ ✖❃ ✣❇ -avoiding permutations containing exactly one 132 sequence is given by ✟ ✧✦ ❃ ✎☞ ✱❃ ✑★ ✪✩ ✬✫ .There are some other recent interesting results.Robertson, Wilf and Zeilberger [32] express the generating function for the number of ❄ ✔❇ ✰❃ -avoiding permutations which have a given number of 123 sequences in form of a continued fraction.Mansour and Vainshtein [27] extend the previous result to determine the generating function of the number of 132-avoiding permutations having a given number of ❄ ✘❃ ✚✁ ✄✁ ✄✁ ✆ sequences.Mansour [25] studies the permutations avoiding a sequence of length four and a nonempty set of sequences of length three.Mansour [26] provides a simple espression for the number of permutations which avoid all the sequences of ✛ ✌✭ ✯✮ ✱✰ ✳✲ ✗✴ ✵✭ ✌✶ ✸✷ ✺✹ ✻✢ (i.e., the sequences of length

✆
having the first element equal to ✹ ).Moreover, he gives a generalization of Robertson's result.
Section 2 of this paper contains the basic definitions about permutations with forbidden subsequences.In Section 3, we describe the tools used to obtain the enumerative results, which are succession rules and generating trees.The former ones consist of rules describing the growing behavior of an object with a fixed parameter value, the latter ones are schematic representations of the former.In Section 4, we express the permutations we are studying in terms of succession rules.We translate the construction, represented by the generating tree, into formulae, thus obtaining a set of functional equations.Their solution gives the generating function of the permutations according to the length, number of left minima, non-inversions and active sites.We are able to determine the generating function according to the length of the permutations, number of left minima and non-inversions.This result allows us to show that the generating function is algebraic of degree two, except for ❅ ✘✷ .

Notations and definitions
In this section we recall the basic definitions about permutations with forbidden subsequences that will be referred to in the next sections.
otherwise it is said to be an inactive site.

Succession rules and generating trees
In this section we briefly describe the tools used to deduce our enumerative results, namely succession rules and generating trees.They were introduced in [12] for the study of Baxter permutations and further applied to the study of permutations with forbidden subsequences by West [11,38,39].Definition 3.1 A generating tree is a rooted, labelled tree such that the labels of the set of children of each node v can be determined from the label of v itself.Thus, any particular generating tree can be specified by a recursive definition consisting of: 1. basis: the label of the root,

inductive step: a set of succession rules that yields a multiset of labelled children depending solely on the label of the parent.
A succession rule contains at least the information about the number of children.Let ✭ be a forbidden subsequence.Following the idea developed in [12], the generating tree for ✭ -avoiding permutations is a rooted tree such that the nodes on level are exactly the elements of ✰ ★ ✟ ✭ ☞ ; the children of a permutation ☞ are all the ✭ -avoiding permutations obtained by inserting ✝ ❄ into ✁ .Labels must be assigned to the nodes and they record the number of children of a given node.
Example 3.1 The Catalan tree and ❄ ✘❃ ✿❇ -avoiding permutations are obtained by the succession rule: The permutation of length one has two active sites (basis in rule The permutation of length one has two active sites (basis in rule with as many as ❅ ✽❄ 's appearing in both above and to the left of this structure. These permutations are enumerated by number sequences such that the -th term is between the -th Catalan number and ✕ (see Figure 2).We describe the structure of their generating tree and use this construction to obtain a set of functional equations satisfied by the generating function of the class of permutations ✏ ✾ .Some computations allow us to determine this generating function according to the length of the permutations, number of left minima and non-inversions.

Related results
The classes of permutations described in this paper are enumerated by numbers lying between the Catalan numbers and the factorial (see Figure 2).The present result is an extension of previously known results for ❅ ✞✷ ❄ and ❅ ✞✷ ❃ .It is easy to prove that the ❃ ✎☞ -th term of the ❅ -th sequence by adding ❅ ✖✕ .
In [11], permutations with the forbidden subsequnces (❄ ✘❃ ✿❇ , (5.5) The parameter ❅ allows us to obtain classes of permutations such that the number of permutations of length is a number lying between ❃ ✑★ ✪✩ ✶ and the -th Catalan number, ★ .The generating functions of these number sequences are all rational, except for ❅ ✗✷ as ❄ ✘❃ ✣❇ -avoiding permutations are enumerated by the Catalan numbers whose generating function is algebraic.
In [3], permutations with the forbidden subsequnces (❇ ✰❃ ❄ , ❄ , are studied.Their generating tree can be expressed in terms of the following succession rule: The parameter ❅ allows us to obtain classes of permutations enumerated by numbers lying between the Motzkin and the Catalan numbers whose generating functions are both algebraic.
In [29], the set of permutations ✰ (5.7) The parameter ❅ allows us to obtain classes of permutations enumerated by numbers lying between the Bell numbers and the factorial both having transcendent generating functions.
Let us note that we describe countably many succession rules which lead to rational, algebraic and transcendent generating functions.These are instances of the general theory developed by C. Banderier et al. in [2].

Fig. 2 :
Fig. 2: First numbers of the sequences counting the permutations in ✦ ✄ .