1. Introduction
The qStirling numbers of the second kind were first studied by Carlitz [4, 5] and Gould[14]. Following Gould[14], we define two related qStirling numbers of the second kind, by recursions as follows,
(1.1) 
and
(1.2) 
where
Note that
In 1978, Milne [18] defined combinatorially by introducing certain statistic on set partitions, which we shall call . Since then, many authors such as Garsia and Remmel [13], Sagan [19], Wachs and White [23], White [26], Ehrenborg and Readdy [9], Wagner [25] have studied statistics with distributions given by or . Most of them were defined on set partitions or restricted growth functions, which are frequently used to encode set partitions.
The concept of restricted growth function was introduced in [15, 27] and the name was coined by Milne [17]. Let . Given a word in , it is said to be a restricted growth function or an if
Wachs and White [23] defined four inversionlike statistics on RGFs, namely, , , and . Let be the set of s of length with , and be the set of all s of length . By Milne[18] and Wachs and White [23], we have the following theorem.
Theorem 1.1.
For , we have
(1.3) 
and
(1.4) 
In this paper, we introduce two new statistics and on s, whose distributions are given by and , respectively.
Theorem 1.2.
For , we have
(1.5) 
and
(1.6) 
Combining (1.3) and (1.5), we see that statistics and have the same distribution on . Combining (1.4) and (1.6), we see that statistics and have the same distribution on . These facts will be proved via a bijection and a bijection , which also preserve some other statistics.
Theorem 1.3.
Theorem 1.4.
Statistics and have the same joint distribution on for all .
The motivation for these two new statistics and stems from their counterparts defined on patternavoiding permutations. We use to denote the set of permutations of avoiding the vincular pattern , and to denote its subset that is composed of permutations with descents. And we use (resp. ) to denote the bigger set with the pattern avoidance condition lifted. As pointed out by Claesson [8, Proposition 3], there is a natural onetoone correspondence between permutations in with descents and set partitions of with blocks, which in turn, can be bijectively mapped to (see for example [20, Theorem 5.1]). Consequently, we have the following three corollaries.
Corollary 1.5.
For , we have
and
Corollary 1.6.
Statistics and have the same joint distribution on for all . (See Table 1 for the case of .)
Corollary 1.7.
Statistics and have the same joint distribution on for all .
We will extend the set of s to the set of unrestricted growth functions, or URGs for short, in order to encode the set of ordered partitions. These are partitions whose blocks can be arbitrarily permuted, in contrast to the “least element increasing” convention for the (unordered) partitions. We take to be the set of URGs of length , and to be the set of all with . Let . Following Steingrímsson [21], the statistics defined on with distribution given by are called EulerMahonian. Steingrímsson introduced a statistic and showed that it is EulerMahonian. Encouraged by Theorem 1.3 and Corollary 1.6, we introduce a new EulerMahonian statistic , defined on URGs.
Theorem 1.8.
For , we have
In the next section, we will first present the formal definitions of all the statistics that concern us here, and the meanings of vincular pattern and , then we go on to explain the transition from Theorems 1.1, 1.2, 1.3 and 1.4 to Corollaries 1.5, 1.6 and 1.7. An algebraic proof of Theorem 1.2 will be given in Section 3. The aforementiond bijection will be constructed in Section 4, to give a proof of Theorem 1.3. While the bijection will be given in Section 5, which leads to a proof of Theorem 1.4. An involution on will be defined in Section 6, to reveal a finer relation between and . Theorem 1.8 will be proved and then strengthened (via the involution ) in Section 7. We conclude with several remarks in the final section.
2. Preliminaries
In this section, we are going to recall or introduce four different but related combinatorial objects, namely, RGFs, patternavoiding permutations, URGs and barred permutations, along with various statistics defined on them.
Given any word (not necessarily an ), we define to be the set of the positions of ascents. We make the convention here, that if is certain coordinate statistic defined for each , then the boldfaced and
stand for the corresponding vector statistic and global statistic, respectively. Wachs and White
[23] gave the following definitions. LetFor , let
As an example, for , we see that
Consequently, we have , , and .
Now we are ready to give descriptions of and , which can be seen as variants of and . Let be the position of the rightmost one in . For , let
For our running example , we have , , , and
It follows that and . In fact, it is evident from the definition that for any word and any , we have
(2.1) 
A less obvious relation is
(2.2) 
As can be checked with the former example, .
An occurrence of a classical pattern in a permutation is a subsequence of that is orderisomorphic to . For example, has two occurrences of the pattern , as witnessed by its subsequences and . is said to avoid if there exists no occurrence of in . In an effort to characterize various Mahonian statistics, Babson and Steingrímsson [2] generalized the notion of permutation patterns, to what are now known as vincular patterns. Adjacent letters in a vincular pattern which are underlined must stay adjacent when they are placed back to the original permutation. For comparison, now contains only one occurrence of the vincular pattern in its subsequence , but not in any more. Recall that a permutation statistic is called Mahonian if it has the same distribution with the number of inversions, denoted , over . Given a vincular pattern and a permutation , we denote by the number of occurrences of the pattern in , and .
Babson and Steingrímsson [2] showed that most of the Mahonian statistics in the literature can be expressed as the sum of vincular pattern functions. We list some of them below.
where stands for the function composition of the reversal and the complement . Given a permutation , recall that
Moreover, let
Actually there exists a stronger relation between the two Mahonian statistics and . In the same paper [2], Babson and Steingrímsson conjectured the bistatistic is EulerMahonian, i.e., it is equidistributed with . This was first proved by Foata and Zeilberger [10], see also [3, 16, 8, 11] for further developements along this line. Since clearly for any permutation , we include here the equivalent version for , which will also be needed in Section 7.
Proposition 2.1 (Theorem 3 in [10]).
For , we have
(2.3) 
The following relation parallels (2.2), and has previously been observed in [12, Lemma 5.4]. For any permutation ,
(2.4) 
Wachs [22] introduced the more general s, to encode the set of ordered partitions. But the following notion of unrestricted growth function seems to be more appropriate for our purpose.
Definition 2.2.
Given a word , it is said to be an unrestricted growth function or a if for any , , appears in .
For a given ordered set partition of , say , we form a word by taking , if and only if . This is clearly seen to be a bijection between ordered partitions of into blocks, and .
Example 2.3.
There are in total six ordered partitions of into two blocks, and also six words in . We list them below in onetoone correspondence, with the first three being the (unordered) ones in .
Besides the total order between all the letters of a given word , imposed by their numerical values, we also need to consider a partial order . By we mean there exists some and , with , and .
Now we can recall the statistics used in [21], reformulated in terms of words, as well as a new statistic block nondescents, denoted by . For any word , we let
where (resp. ) if the statement is true (resp. false).
Note that although for each ,
(2.5) 
in general , for . Take for example, we see , .
As noted by Claesson [8, Proposition 3] or even earlier, there is a natural bijection sending a set partition of with blocks, to a avoiding permutation of with descents. Combining this with the bijection between set partitions and s, we may construct a bijection directly from to that sends to . However, in order to deal with URGs and prove Theorem 1.8, we define our bijection on a bigger set, called the barred permutations, whose definition we give below.
Definition 2.4.
A barred permutation is a permutation decorated with certain amount of bars in between its consecutive letters, according to the following rules.

If , then we must insert a bar between and . We call such bar a fixed bar and use the double line to represent it, although it still counts as one bar for enumeration.

If , then we may or may not insert a bar between and . We call such bar an active bar and use the single line to represent it.
is then called the base permutation of . For , and , we denote by the set of all barred permutations of , and by the subset with exactly active bars and fixed bars.
We note that can be embedded naturally in as the subset , simply by placing a bar at every descent. Consequently, the preimage of each is exactly its base permutation . For this reason, we will use both and interchangeably when it has no active bars.
Given a barred permutation , we now construct a , such that
Example 2.5.
The six s listed in Example 2.3 are reproduced below, as the images of six barred permutations, under the map .
Lemma 2.6.
For all , the map as described above is a bijection between and . It induces a bijection between and . Moreover, for each , suppose , then we have
(2.6) 
Proof.
By our construction, the image of each barred permutation in is clearly in , hence is welldefined. The injectivity of is also clear. The inverse of can be described as reading from left to right, and recording the positions of its largest letters , then putting a bar, next recording the positions of , putting a bar, and so on and so forth. Therefore is indeed a bijection.
When we restrict so that the image set is , then the defining condition for restricted growth functions, namely,
forces to avoid pattern , and to be without any active bars, and vice versa.
In view of (2.6), Corollary 1.5 follows immediately from Theorem 1.1 and Theorem 1.2, while Corollary 1.6 follows immediately from Theorem 1.3 and Corollary 1.7 follows immediately from Theorem 1.4.
The observations (2.2), (2.4), (2.5) and Lemma 2.6, motivated us to generalize the permutation statistic to a statistic for s.
For any , we let
(2.7) 
3. An algebraic proof of Theorem 1.2
In this section, we will give an algebraic proof of Theorem 1.2. The following recurrence given by Wagner [24] will be needed.
Lemma 3.1.
For , we have
(3.1) 
Proof of Theorem 1.2.
We first give a proof of (1.5), which is by induction on . We assume that (1.5) is true for . Given , we consider the following two cases.

.
Assume that . Clearly, is an of length with . By the alternative definition of given in (2.2) and the fact that , it is not hard to see that . Thus, we have

.
Combining the above two cases and (1.1) , we have
This completes the proof of (1.5).
Now we proceed to give a proof of (1.6). Given an , we see that . Let be the obtained from by deleting all the ’s in and decreasing each remaining letter by . Suppose that the number of occurrences of in is . Clearly, is an with length and maximum letter . Conversely, given such a , there are different ways to insert ’s to recover certain .
For an index . If and , then , which equals the contribution to from . If and , then , which equals the contribution to from . The remaining cases can be discussed similarly, which amount to giving
It follows that
By (1.4) in Theorem 1.1, we have
By (3.1), it follows that
This completes the proof of (1.6). ∎
4. A bijective proof of Theorem 1.3
We need two local operators and , which are crucial in the construction of and its inverse . For any word composed of nonnegative integers, take any letter , we define
For any subword of , we abuse the notation and let (resp. ) denote the image of applying (resp. ) on each letter of .
Suppose has the following decomposition, wherein is the rightmost in , is the greatest letter in the prefix of ending at , and , if any, is the leftmost in . It could also be the case that , so no exists.
Note that and are the prefix and suffix of , respectively, and there are no restrictions on the relative orders among . We define our bijection according to the following three cases.

If does not exist, i.e., the prefix of is completely composed of , say , with containing no , then .

If does exist, and or does not exist, then

If both and exist, and , then
It can be easily checked that in all three cases,
(4.1) 
Example 4.1.
In the following two examples, we calculate two triples of statistics that turn out to be the same, via described above.
Comments
There are no comments yet.