1. Introduction
Since the publications of Duncan and Steingrímsson [15], Corteel, Martinez, Savage and Weselcouch [14] and Mansour and Shattuck [24], there has been increasing interest in counting pattern avoiding ascent/inversion sequences [1, 2, 6, 8, 10, 11, 12, 20, 19, 21, 22, 23, 25, 28, 29]. In particular, motivated by the study of generalized patterns in permutations [4, 13], Yan and the first named author [23] carried out the systematic study of ascent/inversion sequences avoiding vincular patterns of length . They reported many nice connections with familiar combinatorial families and posed several challenging enumeration conjectures. The objective of this paper is to address two of their conjectures concerning the pattern in ascent/inversion sequences. It turns out that avoiding ascent and inversion sequences possess attractive enumeration results albeit having elusive structure.
Before stating our results, we need to review some definitions on inversion sequences. An integer sequence of length is an inversion sequence if for all . Inversion sequences of length , denoted , are in natural bijection with permutations of via the famous Lehmer code (see [14, 20]). The set of ascent sequences of length consists of such that
for all , where is the number of ascents of . As one of the most important subsets of inversion sequences, ascent sequences were introduced by BousquetMélou, Claesson, Dukes and Kitaev [9] to encode the +free posets. Many remarkable connections between inversion (resp. ascent) sequences and permutations (resp. restricted permutations) with unexpected applications have been found in the literature; see [27, 20, 17] and the references therein.
Permutations and inversion sequences can both be viewed as words on . A word contains a pattern if there exists such that the subword of is order isomorphic to . In addition, if some consecutive letters in a pattern are underlined, then we further require that in any occurrence of , the letters corresponding to these underlined letters be adjacent in . Such generalized patterns are known as vincular patterns (cf. Kitaev’s book [18, pp.2]), which were introduced in the classification of Mahonian statistics by Babson and Steingrímsson [7]. If a word does not contain an occurrence of a vincular pattern , then is said to avoid the pattern . For example, an inversion sequence is avoiding if there does not exist indices and , , such that . For a set of words, the set of avoiding words in is denoted by .
For and , let
The triangle is known as the triangle of triangular binomial coefficients and appears as A098568 in the OEIS [26]. Lin and Yan [23] proved that enumerates ascent sequences with and conjectured the following different interpretation.
Conjecture 1.1 (Lin and Yan [23, Conj. 3.4]).
For and , we have
Following Yan [28], an ascent sequence is said to be primitive if for all . Let be the set of primitive ascent sequences of length . It was observed in [23] that Conjecture 1.1 is equivalent to
(1.1) 
Alternatively, it suffices to establish the generating function formula
(1.2) 
where the sum runs over all avoiding primitive inversion sequences with ascents and is the length of . Interestingly, the righthand side of (1.2) is also the generating function for domino tilings of Azetec diamond of order by the number of horizontal dominoes, a celebrated result of Elkies, Kuperberg, Larsen and Propp [16].
Another conjecture in [23] concerns an interpretation for the refined powered Catalan numbers in terms of avoiding inversion sequences. The integer sequence of powered Catalan numbers is registered on [26] as A113227, whose first few terms are
It is known that the pattern avoiding classes , , , , and are all counted by the powered Catalan number (see [8, 14, 25] and the references cited therein). The number has a natural refinement by , where are defined recursively by
(1.3) 
Corteel, Martinez, Savage and Weselcouch [14] proved that the cardinality of or is by showing
(1.4) 
where is the number of zero entries of . Lin and Yan [23] showed that has cardinality by establishing a bijection between and but were unable to prove the following refinement.
Conjecture 1.2 (Lin and Yan [23, Conj. 2.20]).
For and , we have
In this paper, we confirm the above two conjectures.
The rest of this paper is organized as follows. In Section 2, we prove Conjecture 1.1 by considering the last entries of avoiding ascent sequences. In Section 3, we prove Conjecture 1.2 via a welldesigned algorithm for constructing avoiding inversion sequences. We will also consider the last entry statistic of avoiding inversion sequences, which leads to a new succession rule for the powered Catalan numbers. Finally, we end this paper with two tempting equidistribution conjectures concerning the open problem to enumerate avoiding permutations.
2. On avoiding ascent sequences
This section is devoted to the proof of Conjecture 1.1. We begin with a different characterization of , which is more convenient for our enumerative purpose.
For a given , since it is primitive, each consecutive pair forms either a descent, or an ascent. Now we can uniquely partition with “”, into maximal decreasing subsequences called runs. Let be the subsequence formed by the least entry in each run of , and we call it the tail sequence of . For example,
We have the following characterization of using tail sequences.
Lemma 2.1.
For any with , we have
Proof.
Clearly, is a consequence of and the definitions of primitive ascent sequences and tail sequences. Now we show that contains a pattern if and only if contains a descent.
Suppose the triple forms a pattern in , then must be the tail of a run. Suppose is the tail of the run that contains , for some . We see , hence , containing and , must have a descent. Conversely, suppose is a descent in , and suppose the tails of the th and th runs, and the largest entry in the th run in the original sequence , are (), () and respectively. Then forms a pattern in , which completes the proof of the lemma. ∎
Let denotes the set of all primitive ascent sequences. For each , define the weight of by , where
If has ascents, then it has exactly tails hence . Therefore, Eq. (1.2) can be rewritten as
(2.1) 
which is equivalent to Conjecture 1.1. In order to prove (2.1), we introduce the refined enumerator
where is the set of all with and the last entry of being . We have the following recursion for .
Lemma 2.2.
For and , we have the recursion
(2.2) 
with the initial conditions , and for .
Proof.
By the characterization in Lemma 2.1, each ascent sequence with tail sequence and the penultimate tail being () is decomposed into

the prefix ,

the entries forming a subset of the interval with the restriction that such a subset must be nonempty whenever (since is primitive), and

the last entry .
Now if we take the weight into consideration, recursion (2.2) follows from the decomposition above immediately. ∎
We are now ready to prove the following expression for .
Theorem 2.3.
For , we have
(2.3) 
Proof of Conjecture 1.1.
3. On avoiding inversion sequences
3.1. Proof of Conjecture 1.2
In this subsection, we develop a delicate algorithm to construct recursively avoiding inversion sequences, which leads to a proof of Conjecture 1.2.
The following operations are quite standard (cf. [14]) for constructing new inversion sequences from old ones. For an inversion sequence and any integer , let
Note that the image is not necessarily an inversion sequence. And sometimes we need to apply on substrings of an inversion sequence. We use concatenation to add an entry to the beginning or the end of an inversion sequence: is the inversion sequence and for , is the inversion sequence . For any sequence , not necessarily an inversion sequence, we use to denote the value of the smallest entry in .
Quite recently, Beaton, Bouvel, Guerrini and Rinaldi [8, Prop. 19] rephrased (1.3) in terms of the following succession rule, and reproved the statement of Corteel et al. for by explaining their growth subjected to this rule.
(3.1) 
Here means copies of . The powered Catalan generating tree (actually an infinite rooted tree) can be constructed from like this: the root is and the children of a vertex labelled are those generated according to the rule . Note that the number of vertices at level that carry the label in the powered Catalan generating tree is precisely the quantity .
Our strategy to prove Conjecture 1.2 is to show that the family also obeys the succession rule . We remark that the first step is the same as given in [8], while the second step involving “Algorithm BS” is substantially different and crucial in dealing with avoiding, rather than avoiding inversion sequences.
Proof of Conjecture 1.2.
For , let . Let and suppose its zero entries are indexed as . Since is avoiding, it uniquely decomposes as
where for , is a nonincreasing, zerofree substring of length , and is a avoiding, zerofree substring of length .
 Step I:

Set
 Step II:

Transform into one or more avoiding inversion sequences, according to the following three succession cases.
 :

Set .
 :

Replace each of by , and denote the new sequence by .
 :

For any , replace each of by . Choose one integer , then go on to replace the zero by , and denote this new sequence by . Apply the following Algorithm BS on . The output sequence is denoted as .
Algorithm BS (backward shift)
Input sequence
where the substrings contain neither nor , and the between and is the only to the left of .
If or , output as is. Note that in this case contains at most one between the s.
Otherwise, initiate and we go through the following steps to locally transform certain substring of .

Find the substring , where or , and is the maximal zerofree substring extended to the left of .

Transform .

If , put and go back to Step 1.
Else if , terminates.
Else put and go back to Step 1.
Output the final sequence.
Example 3.1.
Take with and for example. Applying the succession rules and the algorithm BS gives
where all the involved substrings are colored in red.
For welldefinedness, one checks that following the succession rules , and , we end up respectively, with one sequence , one sequence , and sequences for and .
To complete the proof, we have to show that if we apply the above process for each sequence in , we generate every sequence in precisely once. The first thing to notice is that contains no s, has only one , and has at least two s and at least one . So these three cases are mutually exclusive. It should be clear how to invert or to recover , so it suffices to invert . This is done by first applying the forward shift algorithm below to , which outputs the sequence ; then obtaining from by replacing all s by s; and finally deriving from .
Algorithm FS (forward shift)
Input sequence , which has at least two s and at least one . We call the substring inbetween the leftmost and the rightmost the zero zone of .
If has less than two s in the zero zone, output as is.
Otherwise we can write
where the substrings are zerofree and nonincreasing, is zerofree and avoiding, and (resp. ) contains the leftmost (resp. rightmost) in the zero zone. Now initiate and we go through the following steps to locally transform certain substring of .

Find the substring , where is the maximal zerofree substring extended to the right of .

Transform .

If ends with the rightmost in the zero zone, continue.
If , transform and terminates.
Else transform and terminates.
Else if , put and go back to Step 1.
Else put and go back to Step 1.
Output the final sequence.
Example 3.2.
Take for example. Applying the algorithm FS gives
In conclusion, we have proved that avoiding inversion sequences grow according to the rule if every sequence with zeros is represented by . This completes the proof of the conjecture. ∎
Example 3.3.
In this example, we find all images of an inversion sequence , following the steps described in the proof above.
3.2. The last entry statistic
In this subsection, we study the last entry statistic of inversion sequences and obtain a new succession rule for powered Catalan numbers. For an inversion sequence , let be the last entry of . The last entry statistic has been found to be useful in solving two enumeration conjectures in [19]. By comparing the construction of the rule for avoiding inversion sequences in the proof of Conj. 1.2 and that for avoiding inversion sequences in the proof of [8, Prop. 19], we have the following equidistribution.
Proposition 3.4.
The tripe has the same distribution over and , where is the number of righttoleft minima of an inversion sequence .
Lin and Yan [23, Lem. 2.19] showed that Baril and Vajnovszki’s code [5] restricts to a bijection between and . For a permutation , define the encoding by
The encoding , known as invcode of permutations, is a variation of the famous Lehmer code. One interesting feature of that the code does not possess is , where .
Proposition 3.5.
The invcode restricts to a bijection between and . Consequently, the triple over (or ) is equidistributed with over , where (resp. ) denotes the number of lefttoright maxima (resp. righttoleft maxima) of a permutation .
Proof.
Let and . If contains the pattern , then there exists such that and for each . Thus, we have and so forms a pattern in . Conversely, suppose that and is a pattern of . Since , we have and there exists such that . Now forms a pattern in . This completes the proof. ∎
In [8, Prop. 25], Beaton, Bouvel, Guerrini and Rinaldi obtained another succession rule for the powered Catalan numbers, which is essentially different from :
The consideration of the last entry statistic on avoiding inversion sequences leads to a third succession rule for the powered Catalan numbers.
For a sequence , let us introduce the parameters of by
and
Proposition 3.6.
The avoiding inversion sequences grow according to the following succession rule
Proof.
Let be a sequence in with parameters . It is clear that the sequence is in if and only if , where equals the largest ascent bottom of . We consider two cases:

If , then forms an ascent of whose ascent bottom is obviously not smaller than . So if we write for some , then the parameters of are .

If , then for some . In this case, the parameters of are .
Summing over all the above two cases results in the succession rule . ∎
4. Two equidistribution conjectures
The classification of Wilf equivalences for vincular patterns of length in inversion sequences has been completed, thanks to Auli and Elizalde’s recent work^{1}^{1}1Auli and Elizalde independently initiated their work, we thank them for keeping us informed. [3]. Towards the complete classification of vincular patterns of length in permutations, Baxter and Shattuck conjectured [7] that has cardinality , the th powered Catalan number. In their attempt to prove this conjecture, Beaton, Bouvel, Guerrini and Rinaldi [8, Conj. 23] found the following refinement.
Conjecture 4.1.
The number of permutations of with righttoleft minima is .
Conjecture 4.1 is equivalent to the assertion that the statistic ‘’ over or has the same distribution as ‘’ over , where denotes the number of righttoleft minima of a permutation . Using Maple program, we find the following refinement of Conjecture 4.1.
Conjecture 4.2.
The quadruple on has the same distribution as on .
Here we use (resp. ) to denote the number of lefttoright minima (resp. ascents) of a permutation . And for an inversion sequence , the two statistics involved are
Conjecture 4.2 has been verified for .
Finally, the consideration of the last entry statistic leads to another refinement of Baxter and Shattuck’s enumeration conjecture.
Conjecture 4.3.
The pair on
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