1. Introduction
An occurrence of a classical pattern in a permutation is a subsequence of that is orderisomorphic to , i.e., that has the same pairwise comparisons as . For example, has two occurrences of the pattern in its subsequences and . The study of (classical) permutation patterns have mostly been with respect to avoidance, i.e., on the enumeration of permutations in the symmetric group that have no occurrences of the pattern(s) in question. The tone is generally believed to be set in 1968, when Knuth published Volume one of The Art of Computer Programming [12]. It was observed in [12] that the number of avoiding permutations are enumerated by the Catalan numbers. See [10] and the references therein for a comprehensive introduction to patterns in permutations.
In 2000, Babson and Steingrímsson [2] traced out a new line of research by generalizing classical patterns to what are now known as vincular patterns. In an occurrence of a vincular pattern, some letters of that orderisomorphic subsequence may be required to be adjacent in the permutation, and this is done by underscoring the adjacent letters in the expression for the pattern. For our previous example, only represents an occurrence of the vincular pattern in . As was investigated thoroughly in [2], these vincular patterns turned out to be the “building blocks” in their classification of Mahonian statistics. Recall that a permutation statistic is called Mahonian precisely when it has the same distribution, on , as , the number of inversions. Given a vincular pattern and a permutation , we denote by the number of occurrences of the pattern in . When it causes no confusion, we will even drop the parentheses. For instance, it was observed/defined in [2]:
(1.1) 
Note that the first three Mahonian statistics are wellknown ones putting on a new look, while the last one was one of those introduced in [2] and shown to be Mahonian.
Now we state the joint equidistribution results involving STAT obtained by Burstein [3] on and by Kitaev and the third author [11] on , the set of length words over the alphabet . These two results largely motivated this work. All undefined statistics will be introduced in the next section. For the spelling of the statistics, we follow the convention of [6], i.e., all Mahonian statistics are spelled with uppercase letters, all Eulerian statistics are spelled with lowercase letters, while all the remaining ones are merely capitalized. Moreover, the italic letters are used for setvalued statistics, such as Id in Theorem 1.3 and Corollary 1.5.
Theorem 1.1 (Theorem 2.1 in [3]).
Theorem 1.2 (Theorem 2 in [11]).
Statistics and
have the same joint distribution on for all and .
A quick look at Table 1 reveals that when one wants to extend the above results to the rearrangement class , the statistic must be dropped. In the meantime, the involution we are going to construct actually preserves the inverse descent set Id (see Definition 2.2). The following are the main results of this paper.
Theorem 1.3.
Statistics and
have the same joint distribution on for all .
Remark 1.4.
It is worth mentioning that in [7] it has been shown that and are equidistributed on for all .
In view of this theorem and the standard “coding/decoding” between words and permutations [6], we derive the following version for words, where the statistics have to be redefined on words accordingly. We defer the details to the next section.
Corollary 1.5.
Statistics and
have the same joint distribution on for any word .
Since uniquely determinates both and (see Definition 2.2), the next corollary is a consequence of the previous one, and it is similar to Theorem 1.2, except is replaced by and the equidistribution is stated on the rearrangement classes.
Corollary 1.6.
Statistics and
have the same joint distribution on for any word .
In the next section, we will first present the formal definitions of all the statistics that concern us here, and then explain the transition from Theorem 1.3 to Corollary 1.5. Theorem 1.3 will be proved in Section 3, where FoataSchützenberger’s involution via RSK correspondence will be recalled. We conclude with some further questions suggested by this work.
1122  0  0  0  1  0  0  0 

1212  1  1  1  1  2  2  3 
1221  0  1  1  1  2  3  2 
2112  0  1  1  2  2  1  2 
2121  0  2  1  2  2  4  4 
2211  0  1  1  2  2  2  1 
2. Preliminaries
Since all the statistics that concern us here can be defined on both permutations and words, and that we have two versions of joint equidistribution results, some care needs to be taken to avoid unnecessary repetitions. We decide to define all the statistics on the rearrangement class for a word of length , making sure that when is composed of distinct letters, and thus becomes essentially , all the statistics reduce to the original ones defined on . On the other hand, for the two equidistribution results we first explain how to derive Corollary 1.5 from Theorem 1.3 and then construct, in the next section, the key involution on .
We consider words on a totally ordered alphabet . Without loss of generality, we can always take to be the interval . By rearrangement class , we mean the set of all words that can be obtained by permuting the letters of . Within each class , there is a unique nondecreasing word that we denote by . If the letters of are distinct, then is in bijection with the symmetric group via the following “coding” map [6].
Definition 2.1.
Given a word , we code it by the permutation uniquely defined by
This gives rise to a map , which we call the coding map, and clearly is a surjection if and only if .
For example, the word is coded by . Now we are ready to define all seven statistics encountered in the introduction. The true or false function is used in the definition of for our convenience: (resp. ) if the statement is true (resp. false).
Definition 2.2.
Let be a word. The descent set of , , the inverse descent set of , , and a peculiar index set (needed in Section 3) that we call the shuffle set of , are setvalued statistics defined as follows:
We define the following seven statistics on :
We note that all have been extended to words in [11]. Some of the definitions given here may look different but essentially are equivalent to those used in [11], and our formulation of makes it more convenient to define on words.
When the word happens to be a permutation in , then and all the above statistics agree with their original counterparts defined on permutations. This hopefully justifies our abuse of notation, i.e., we use the same name for these statistics, both on words and permutations. Moreover, it should be routine to verify that the coding map preserves six statistics defined above, with being the only exception.
(2.1) 
For the previous example with and , we see , , , etc.
In order to deduce Corollary 1.5 from Theorem 1.3, we need to pay some attention to the “decoding” map from back to . In general, the preimage of a given permutation under the coding map is not unique and therefore this decoding is not well defined. For instance, for both and , we see , with . This should be somewhat expected since when contains repeated letters, we have . To deal with this, we introduce the notion of compatible permutations.
Definition 2.3.
Given the rearrangement class for a word of length , we denote the subset of that is the image of under the coding map , and we call it the set of compatible permutations with respect to . More precisely,
It is clear from Definition 2.1 that if we restrict on a rearrangement class , it is injective, hence it induces a bijection between and , with a unique inverse defined on :
(2.2)  
(2.3) 
The next proposition characterizes using and the inverse descent set.
Proposition 2.4.
Given and suppose , where and
Then if and only if .
Proof.
Take a permutation , then for some , and consequently , this takes care of the “only if” part. We finish the proof by simply noting that both and are enumerated by . ∎
In the next section, we are going to construct an involution on such that for any ,
(2.4) 
This proves Theorem 1.3. Now consider the function composition
Note that for any , , and since , we have thanks to Proposition 2.4, thus is welldefined. Now we show that is an involution on that proves Corollary 1.5.
3. Involution
In this section we aim to construct the aforementioned map from into itself, and we show that i) it is an involution and ii) it satisfies (2.4). First off, we recall an involution defined on that was first considered by Schützenberger [13] and further explored by FoataSchützenberger [7]. It builds on two permutation symmetries, namely reversal r and complement c, as well as the famous RobinsonSchenstedKnuth (RSK) algorithm [14, 7.11]. We collect the relevant definitions and notations below. Given a permutation ,
where is a bijection between
and the set of ordered pairs of
cell standard Young tableaux (SYT) with the same shape. Note that , and are involutions on .We usually call the insertion tableau and the recording tableau of . In general, for two unrelated permutations , and are unlikely to be of the same shape. However, it was noted in [7, Prop. 5.2] that and are indeed of the same shape, so the following map is a welldefined involution on .
The next property of is of great importance for our later use and is the main reason that urged us to involve in our construction of .
Lemma 3.1 (Theorem 2 in [7]).
For any , the map as defined above preserves the inverse descent set and exchanges the descent set with its complement to . In other words one has simultaneously:
(3.1)  
(3.2) 
Now given a permutation with , let us denote as (resp. ) the top (resp. bottom) subword that is composed of all the letters larger (resp. smaller) than . We also need the shuffle set defined in Definition 2.2. For example, let , then , and . Conversely, it should be clear how to recover the unique permutation that corresponds to an appropriate triple , so we omit the details. We are ready to describe our key map using this defining triple: (top subword, bottom subword, shuffle set).
Definition 3.2.
Given , let be the unique permutation that corresponds to , where
(3.3)  
(3.4)  
(3.5) 
Remark 3.3.
It should be noted that the idea to associate with the triple is motivated by Burstein’s involution introduced in [3] to prove Theorem 1.1. Actually in our setting, Burstern’s can be defined using the same (3.5), together with (3.3) and (3.4) replaced by
Moreover, in view of (3.1) and Proposition 2.4, (3.3) is welldefined. Thirdly, by the above definition, it follows that . Lastly, using the fact that is an involution, it is routine (using the associated triple) to check that for any , i.e., is an involution on as claimed.
All remained to be done is to prove (2.4). Namely, we need to show for every :
We break it up into the following three lemmas and one corollary.
Lemma 3.4.
Proof.
First note that when the first letter is fixed, only depends on and , and is not affected by at all. Then we simply use (3.1) to finish the proof. ∎
Lemma 3.5.
.
Proof.
This is precisely Lemma 2.7 in [3], hence we omit the proof. ∎
Lemma 3.6.
.
Proof.
The basic idea is to pair the descents of with the descents of . We discuss by two cases.

If , then , and according to (3.5), , where is the length of the maximal block of consecutive entries of that ends with and is contained in . Therefore this pair contributes
to the sum .
All such as defined above clearly sum up to the total length of :
We add the contributions from both cases to arrive at the desired identity. ∎
Corollary 3.7.
Proof.
This involution is admittedly a bit involved, so we include one complete example here to illustrate it.
Example 3.8.
Given , take . We first compute the statistics on :
Now let , we proceed to find . First we get the defining triple associated with :
Applying and according to (3.5), we get immediately . But for and , which involve the map , we need to invoke the RSK algorithm. We first apply RSK on and to get the following four SYT. 3+1 111 122 133 214 0.81.6 3+1 111 122 133 214 0.81.6 3+1 111 123 134 212 0.81.6 3+1 111 123 134 212 0.81.6 Then we apply inverse RSK on the pair to get , hence . Similarly for , we apply RSK on and to get the following four SYT. 2+1+1 111 122 213 314 0.62 2+1+1 111 124 212 313 0.62 2+1+1 111 124 212 313 0.62 2+1+1 111 122 213 314 0.62 Reversing RSK on then produces . Together we have
which corresponds uniquely to . Therefore . Finally we check the statistics match up indeed (with and switched).
4. Further questions
Seeing our equidistribution result in Corollary 1.5, a natural next step is to consider the socalled “extension” (see [8, 9, 5]). Namely, for our prototype of the alphabet, , we take two nonnegative integers and such that . Then we call the letters small and the letters large. Now consider any total ordering, say , on that is compatible with , in the sense that for any small letter and any large letter , we have . Both and have their extensions defined with respect to , so one may ask if there exists a extension for , such that the joint distribution of are still the same as that of .
Finally, Corollary 1.5 justifies as a new EulerMahonian (over rearrangement class of words) pair in the family consisting of and (see [6] for undefined statistics and further details). It would be interesting to pair other Mahonian statistics discovered in [2] with either or , and then pigeonhole the resulting pairs into the existing (or even new) families of EulerMahonian statistics. Another possible direction for furthering the study on is to consider the equidistribution problems on pattern avoiding permutations, see the recent work in [1, 4].
Acknowledgement
The first two authors were supported by the Fundamental Research Funds for the Central Universities (No. CQDXWL2014Z004) and the National Natural Science Foundation of China (No. 11501061).
References
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