Equidistributions of Mahonian statistics over pattern avoiding permutations

A Mahonian d-function is a Mahonian statistic that can be expressed as a linear combination of vincular pattern statistics of length at most d. Babson and Steingrimsson classified all Mahonian 3-functions up to trivial bijections and identified many of them with well-known Mahonian statistics in the literature. We prove a host of Mahonian 3-function equidistributions over pattern avoiding sets of permutations. Tools used include block decomposition, Dyck paths and generating functions.


Introduction
A combinatorial statistic on a set S is a map stat : S → N. The distribution of stat over S is given by the coefficients of the generating function σ∈S q stat(σ) . Let S n be the set of permutations σ = a 1 a 2 · · · a n of the letters [n] = {1, 2, . . . , n} and let σ(k) denote the entry a k . Let S = n≥0 S n . The inversion set of σ ∈ S n is defined by Inv(σ) = {(i, j) : i < j and σ(i) > σ(j)}. A particularly well-studied statistic on S n is inv : S n → N, given by inv(σ) = | Inv(σ)|. An elegant formula for the distribution of the inversion statistic was found in 1839 by Rodrigues [25] σ∈Sn q inv(σ) = [n] q !, where [n] q ! = [1] q [2] q · · · [n] q and [n] q = 1 + q + q 2 + · · · + q n−1 . The descent set of σ is defined by Des(σ) = {i : σ(i) > σ(i + 1)}. In 1915 MacMahon [23] showed that inv has the same distribution as another statistic, now called the major index (due to MacMahon's profession as a major in the british army), given by maj(σ) = i∈Des(σ) i. We also write imaj(σ) = maj(σ −1 ). In honor of MacMahon any permutation statistic with the same distribution as maj is called Mahonian. Mahonian statistics are well-studied in the literature. Since MacMahon's initial work many new Mahonian statistics have been identified. Babson and Steingrímsson [1] showed that almost all (at the time) known Mahonian statistics can be expressed as linear combinations of statistics counting occurrences of vincular patterns. They further made several conjectures regarding new vincular-pattern based Mahonian statistics. These have since been proved and reproved at various levels of refinement by a number of authors (see e.g. [16,4,31,6]).
Two distinct sequences of integers a 1 a 2 · · · a n and b 1 b 2 · · · b n are said to be order isomorphic provided a i < a j if and only if b i < b j for all 1 ≤ i < j ≤ n. A vincular pattern (also known as generalized pattern) of length m is a pair (π, X) where π is a permutation in S m and X ⊆ {0} ∪ [m] is a set of adjacencies. Notationwise, adjacencies are indicated by underlining the adjacent entries in π (see Example 1.1). If 0 ∈ X (m ∈ X), then we denote this by adding a square bracket at the beginning 1 (end) of the pattern π. If X = ∅, then (π, X) coincides with the definition of a classical pattern. A permutation σ = a 1 a 2 · · · a n ∈ S n contains the vincular pattern (π, X) if there is a m-tuple 1 ≤ i 1 ≤ i 2 ≤ · · · ≤ i m ≤ n such that the following three criteria are satisfied • a i1 a i2 · · · a im is order-isomorphic to π, • i j+1 = i j + 1 for each j ∈ X \ {0, m} and • i 1 = 1 if 0 ∈ X and i m = n if m ∈ X. We also say that a i1 a i2 · · · a im is an occurrence of π in σ. We say that σ avoids π if σ contains no occurrences of π. We denote the set of permutations in S n avoiding the pattern π by S n (π). Moreover if Π is a set of patterns, then we set S n (Π) = π∈Π S n (π).
In this paper we shall also need an additional generalization of vincular patterns, allowing us to restrict occurrences to particular value requirements. Let υ = (υ 1 , . . . , υ m ) where υ i ∈ N ⊔ {-}. Define a value restricted vincular pattern to be a triple (π, X, υ) where (π, X) is a vincular pattern. We use the notation (π, X) υ to denote such a pattern. We say that a i1 a i2 · · · a im is an occurrence of (π, X) υ in σ if it is an occurrence of the vincular pattern (π, X) and a ij = υ j whenever υ j ∈ N for j = 1, . . . , m. Note in particular that (π, X) (-,...,-) = (π, X).
In this paper we mainly study equidistributions of the form where Π 1 , Π 2 are sets of patterns and stat 1 , stat 2 are permutation statistics. We will almost exclusively focus on the case where Π i consists of a single classical pattern of length three and stat i is a Mahonian statistic. Although Mahonian statistics are equidistributed over S n , they need not be equidistributed over pattern avoiding sets of permutations. For instance maj and inv are not equidistributed over S n (π) for any classical pattern π ∈ S 3 . Neither do the existing bijections in the literature for proving equidistribution over S n necessarily restrict to bijections over S n (π). Therefore whenever such an equidistribution is present, we must usually seek a new bijection which simultaneously preserves statistic and pattern avoidance. Another motivation for studying equidistributions over permutations avoiding a classical pattern of length three, is that |S n (π)| = C n for all π ∈ S 3 where C n = 1 n+1 2n n is the nth Catalan number (see [20]). Therefore equidistributions of this kind induce equidistributions between statistics on other Catalan objects (and vice versa) whenever we have bijections where the statistics translate in an appropriate fashion. We prove several results in this vein where an exchange between statistics on S n (π), Dyck paths and polyominoes takes place. In general, studying the generating function (1.1) provides a rich source of interesting q-analogues to well-known sequences enumerated by pattern avoidance and raises new questions about the coefficients of such polynomials.
Equidistributions such as (1.1) have been studied in the past. For instance, Burstein and Elizalde proved the following result involving the Mahonian Denert statistic where Exc(σ) = (σ(i)) σ(i)>i and NExc(σ) = (σ(i)) σ(i)≤i . In particular it was shown in [13] that I n (132; q) = I n (213; q) = C n (q) and I n (231; q) = I n (312; q) =C n (q) where The polynomial C n (q) is known as the Carlitz-Riordan q-analogue of the Catalan numbers and have been studied by numerous authors (though no explicit formula is known). Similar recursions for maj have been studied in [13,7].
To decompose pattern avoiding permutations we will require some effective notation. Given permutations τ ∈ S k and σ 1 , σ 2 . . . , σ k ∈ S, the inflation of τ by σ 1 , σ 2 . . . , σ k is the permutation τ [σ 1 , σ 2 , . . . , σ k ] obtained by replacing each entry τ (i) by a block of length |σ i | order isomorphic to σ i for i = 1, . . . , k such that the blocks are externally order-isomorphic to τ . Let σ ∈ S n . Recall that the descent set of σ is given by Des(σ) = {i : σ(i) > σ(i + 1)}. The set of descent bottoms and descent tops of σ are given respectively by DB(σ) = {σ(i + 1) : i ∈ Des(σ)} and DT(σ) = {σ(i) : i ∈ Des(σ)}. Likewise the ascent set of σ is given by Asc(σ) = {i : σ(i) < σ(i + 1)} and we define the set of ascent bottoms and ascent tops of σ to be AB(σ) = {σ(i) : i ∈ Asc(σ)} and AT(σ) = {σ(i + 1) : i ∈ Asc(σ)} respectively. An entry σ(j) is called a left-to-right maxima if σ(j) > σ(i) for all i < j. Let LRMax(σ) denote the set of left-to-right maxima in σ and let lrmax(σ) = | LRMax(σ)|. Similarly an entry σ(j) is called a left-to-right minima if σ(j) < σ(i) for all i < j. Let LRMin(σ) denote the set of left-to-right minima in σ and let lrmin(σ) = | LRMin(σ)|. We call σ If σ = a 1 a 2 · · · a n−1 a n , then the reverse of σ is given by σ r = a n a n−1 · · · a 2 a 1 and the complement of σ by σ c = (n − a 1 + 1)(n − a 2 + 1) · · · (n − a n−1 + 1)(n − a n + 1). The inverse of σ (in the group theoretical sense) is denoted by σ −1 . The operations complement, reverse and inverse are often referred to as trivial bijections and together they generate a group isomorphic to the Dihedral group D 4 of order 8 acting on S n . If π is a classical pattern and g ∈ D 4 , then it is not difficult to see that σ ∈ S n (π) if and only if σ g ∈ S n (π g ). However if π is a non-classical pattern, then there is no such corresponding statement for inverse. Therefore taking the inverse should not be viewed as a 'trivial bijection' in the same sense as complement and reverse when it comes to vincular patterns.
In Table 1 we list the vincular pattern specifications of the Mahonian statistics that we shall consider from [1]. See the references in Table 1 for the original definitions of these statistics. According to [1], Table 1 is the complete list of Mahonian 3-functions (up to trivial bijections) i.e. Mahonian statistics that can be written as a sum of vincular pattern statistics of length at most three. Since some of these statistics have received no conventional name in the literature we will take the liberty of naming them according to the initials of the authors who first proved their Mahonity.

Equidistributions via block decomposition
The equidistributions proved in this section are shown by directly exhibiting a bijection. The bijections are based on standard decompositions of pattern avoiding permutations, or rely on specifying data by which pattern avoiding permutations are uniquely determined. In many cases we are able to find a more refined equidistribution. We begin by proving that maj and mak are related via the inverse map over certain pattern avoiding sets of permutations. This may seem unexpected given that vincular patterns do not behave as straightforwardly under the inverse map as they do under complement and reverse.
The right hand side of (2.1) is known as MacMahon's q-analogue of the Catalan numbers [24].
The following lemma regarding the structure of S n (321) is part of folklore pattern avoidance (see e.g. [20]).  Proof. Let σ ∈ S n (321). By Lemma 2.3 we may decompose σ as where u 1 , . . . , u t are non-empty factors of left-to-right maxima in σ and v 1 , . . . , v t are non-empty factors (except possibly v t ) such that v 1 v 2 · · · v t is an increasing subword. Assume first that v t = ∅. Let M i = max(u i ) and We now define an involution such that maj(φ(σ)) = mak(σ), preserving all pairs of descent top and descent bottoms. For convenience, set M 0 = −∞ and M t+1 = ∞. Let u ′ k denote the unique increasing word of the letters in the set with M k adjoined at the end and let v ′ k denote the unique increasing word of the letters in the set {β ∈ū : m k < β < M k+1 } , with m k adjoined at the beginning for k = 1, . . . , t. Define Thus φ effectively swapsū = LRMax(σ) \ DT(σ) withv = [n] \ (LRMax(σ) ∪ DB(σ)) (when v t = ∅) and DB(φ(σ)) = DB(σ), DT(φ(σ)) = DT(σ). Hence φ is an involution. We have since under the involution φ, each β ∈ LRMax(σ) \ DT(σ) precisely passes the number of descent bottoms that are less than it to its right. Therefore β is involved in the same number of 231 occurrences in σ as φ(β) is involved in 312 occurrences in φ(σ). Hence The statement is proved analogously over S(123).
Hence [132] mak = [312] mak . The remaining mak-Wilf equivalence is proved similarly invoking Proposition 2.1. The inequivalences between the four classes is easily verified by hand or with computer.
Remark 2.6. The charge statistic is also a Mahonian statistic related to maj via trivial bijections by maj(σ) = charge(((σ r ) c ) −1 ) (see [18]). It is worth noting that the mak-Wilf classes in Proposition 2.5 coincide with the charge-Wilf classes identified in [18].
Remark 2.7. It can be checked that maj, inv and mak are the only statistics in Table 1 with non-singleton st-Wilf classes for single classical patterns of length three.
The bijection (2.2) in Theorem 2.4 induces an interesting equidistribution on shortened polyominoes. A shortened polyomino is a pair (P, Q) of N (north), E (east) lattice paths P = (P i ) n i=1 and Q = (Q i ) n i=1 satisfying (i) P and Q begin at the same vertex and end at the same vertex. (ii) P stays weakly above Q and the two paths can share E-steps but not N -steps.
Denote the set of shortened polyominoes with |P | = |Q| = n by H n . For (P, Q) ∈ H n , let Proj Q P (i) denote the step j ∈ [n] of P that is the projection of the i th step of Q on P . Let Proof. We begin by recalling a bijection Υ : H n → S n (321) due to Cheng-Eu-Fu [8]. Given (P, Q) ∈ H n , set Label P (i) = i and Label Q (i) = Label P (Proj Q P (i)). Then Υ(P, Q) = Label Q (1) · · · Label Q (n) ∈ S n (321) is a bijection.
Below we provide a brief account for a well-known lemma due to Simion and Schmidt which will be used to justify the bijection in the next theorem. Proof. It is clear that the left-to-right minima are positioned in decreasing order relative to each other. Now fill in the remaining numbers from left to right, for each empty position i choosing the smallest remaining entry that is larger than the closest left-to-right minima m in position before i. If the remaining numbers are not entered in this unique way and y is placed before x where y > x, then myx is an occurrence of the pattern 132.
Theorem 2.10. For any n ≥ 1, Indeed if σ(i) ∈ DB(σ) and σ(j) < σ(i) for some j < i, then σ(j)σ(i − 1)σ(i) is an occurrence of 132. Hence by Lemma 2.9 we have that σ is uniquely determined equivalently by its first letter, Des(σ) and DB(σ). We define a map φ : S n (132) → S n (132) by requiring We claim that a permutation φ(σ) ∈ S n (132) with the above requirements exists. If the claim holds, then the image of σ is uniquely determined by the data above and therefore φ is well-defined. It also immediately follows that φ is a bijection.
Let i 1 < · · · < i m be the descents of σ. Suppose To show that φ is well-defined we show that the insertion procedure from Lemma 2.9 is always valid. Given a descent bottom (i.e. left-to-right minima) σ(i k + 1) in position n − σ(i j k ) + 2 we must show that there exists enough remaining numbers greater than σ(i k + 1) to fill in the gap to the next descent bottom σ(i k+1 + 1). Within the filling procedure, next after the descent bottom σ(i k + 1), there exists numbers remaining that are greater than σ(i k + 1). There are positions to fill in the gap between the descent bottoms σ(i k + 1) and σ(i k+1 + 1). By minimality so there are enough numbers remaining to fill in the gap. Hence φ is well-defined. Finally, Since also φ(LRMin(σ)) = LRMin(σ), the theorem follows.
Below we provide an additional list of information uniquely determining permutations in S n (231).
where p i and v i are peaks resp. valleys and a i and d i are (possibly empty) increasing resp. decreasing words for i = 1, . . . , m.
We claim that the pairs in P are relatively positioned in increasing order of the valleys. Indeed let (p, v), This in turn implies that v ′ pv is an occurrence of 231 giving a contradiction. Therefore (p, v) is ordered before (p ′ , v ′ ) proving the claim.
Next we claim that the decreasing words d j are uniquely determined. Going from right to left, let d j be the unique decreasing word of all remaining letters (in value) between p j and v j for j = m, . . . , 1. If we do not insert the letters this way and v j < σ i < p j , where σ i is positioned before p j (and hence v j ) then σ i p j v j is an occurrence of 231 which is forbidden.
Since there are no occurrences of 231 in σ by assumption, the letters between each pair of descent top and descent bottom occur to the right of the pair. Therefore the number of occurrences of 312 in σ is given precisely by On the other hand note that ((213) + (321) + (21))σ = α∈DT(σ) (n − α + 1). Thus Finally since des(φ(σ)) = des(σ), the theorem follows.
Remark 2.13. Via Proposition 2.1 we may deduce further equidistributions between maj and foze, see Table 2 in §5 for a summary.

Equidistributions via Dyck paths
A Dyck path of length 2n is a lattice path in Z 2 between (0, 0) and (2n, 0) consisting of up-steps (1, 1) and down-steps (1, −1) which never go below the xaxis. For convenience we denote the up-steps by U and the down-steps by D enabling us to encode a Dyck path as a Dyck word (we will refer to the two notions interchangeably). Let D n denote the set of all Dyck paths of length 2n and set D = n≥0 D n . For P ∈ D n , let |P | = 2n denote the length of P . There are many statistics associated with Dyck paths in the literature. Here we will consider several Dyck path statistics that are intimately related with the inv statistic on pattern avoiding permutations.
Let P = s 1 · · · s 2n ∈ D n . A double rise in P is a subword U U and a double fall in P a subword DD. Let dr(P ) and df(P ) respectively denote the number of double rises and double falls in P . A peak in P is an up-step followed by a down-step, in other words, a subword of the form U D. Let Peak(P ) = {p : s p s p+1 = U D} denote the set of indices of the peaks in P and npea(P ) = | Peak(P )|. For p ∈ Peak(P ) define the position of p, pos P (p), resp. the height of p, ht P (p), to be the x resp. y-coordinate of its highest point. A valley in P is a down step followed by an up step, in other words, a subword of the form DU . Let Valley(P ) = {v : s v s v+1 = DU } denote the set of indices of the valleys in P and nval(P ) = | Valley(P )|. For v ∈ Valley(P ) define the position of v, pos P (v), resp. the height of v, ht P (v), to be the x resp. y-coordinate of its lowest point. For each v ∈ Valley(P ), there is a corresponding tunnel which is the subword s i · · · s v of P where i is the step after the first intersection of P with the line y = ht P (v) to the left of step v (see Figure 2). The length, v − i, of a tunnel is always an even number. Let Tunnel(P ) = {(i, j) : s i · · · s j tunnel in P } denote the set of pairs of beginning and end indices of the tunnels in P . Cheng et.al. [7] define the statistics sumpeaks and Define the area of P , denoted area(P ), to be the number of complete √ 2 × √ 2 tiles that fit between P and the x-axis (cf [19]). Burstein and Elizalde [5] define a statistic which they call the 'mass' of P . We will define two versions of it, one pertaining to the U -steps and one to the D-steps. For each i ∈ Up(P ) define the mass of i, mass P (i), as follows. If s i+1 = D, then mass P (i) = 0. If s i+1 = U , then P has a subword of the form s i U P 1 DP 2 D where P 1 , P 2 are Dyck paths and we define mass P (i) = |P 2 |/2. In other words, the mass is half the number of steps between the matching D-steps of two consecutive U -steps. The part of the Dyck path P contributing to the mass of each of the first three U -steps is highlighted with matching colours in Figure 2. Define mass U (P ) = i∈Up(P ) mass P (i).
The statistic mass U coincides with the mass statistic defined by Burstein and Elizalde [5]. Analogously if i ∈ Down(P ), define mass P (i) = 0 if s i−1 = U . If s i−1 = D, then P has a subword of the form U P 1 U P 2 Ds i where P 1 , P 2 are Dyck paths and we define mass D (s) = |P 1 |/2. In other words, the mass is half the number of steps between the matching U -steps of two consecutive D-steps. Define mass D (P ) = i∈Down(P ) mass P (i).
We now define another well-known map Γ : S n (321) → D n due to Krattenthaler [21] which also appears in a slightly different form in the work of Elizalde [14]. Let σ ∈ S n (321) and consider an n × n array with crosses in positions (i, π i ) for 1 ≤ i ≤ n, where the first coordinate is the column number, increasing from left to right, and the second coordinate is the row number, increasing from bottom to top. Consider the path with north and east steps from the lower-left corner to the upper-right corner of the array, whose right turns occur at the crosses (i, σ i ) with σ i ≥ i. Define Γ(σ) to be the Dyck path obtained from this path by reading a U -step for every north step and a D-step for every east step of the path. The bijection is illustrated in Figure 3.
Next we define a Dyck path bijection Ψ : D n → D n due to Cheng et.al. [7] that is weight preserving between the statistics spea and stun.
First we define a bijection δ : n−1 k=0 D k × D n−k−1 → D n as follows. Given two Dyck paths where all exponents are positive, define δ(Q, R) by if R = ∅. If Q = ∅ the same definition works with the convention that a 1 = b 1 = 0.
We will now interpret mad over both S n (231) and S n (312) in terms of Dyck path statistics under ∆. The following theorem is a straightforward modification of Theorem 3.11 in [5].
If P = P 1 · · · P r where P i is a Dyck path returning to the x-axis for the first time at its endpoint, then define Θ(P ) = Θ(P 1 ) · · · Θ(P r ). Assume therefore r = 1 and write The map Θ is clearly a bijection. Note that Hence by induction it follows that mass U (Θ(P )) + dr(Θ(P )) = sups(P ).
The following corollary answers a question of Burstein and Elizalde in [5].

Proof. Combine Theorem 3.4 (ii) with (3.2).
Below we find an interpretation of Theorem 1.1 in terms of Dyck path statistics.
The first statement in the proposition follows from Theorem 3.10 and a similar observation to above.
Remark 3.12. By Theorem 1.1 the Dyck path statistics in Proposition 3.11 are equidistributed over D n .

Equidistributions via generating functions
In this section we use generating functions to derive equidistributions (albeit nonbijectively) between Mahonian statistics over S n (π). We also provide a recursion for a more general statistic involving arbitrary linear combinations of vincular pattern statistics of length three. This recursion generalizes for instance the recursions in [13]. .
Putting t = 1 and t = q 2 , eliminating F (q, q 2 , z) from the resulting equation system and solving for F (q, 1, z) we obtain the continued fraction expansion in the theorem.
A similar argument for rsist over S(312) gives a matching continued fraction expansion. We leave the details to the reader.
Using almost identical arguments to Theorem 4.1 we may moreover prove the following equidistributions. By combining Theorem 4.1 and Theorem 4.4 with Theorem 3.8 and Corollary 3.6 we may deduce further equidistributions between inv and the statistics foze ′ , sist, sist ′ and sist ′′ , see Table 2 in §5 for a summary.
For each k ≥ 1, let ι k−1 = (12 · · · k) denote the statistic that counts the number of increasing subsequences of length k in a permutation. Define ι −1 by ι −1 (σ) = 1 for all σ ∈ S (i.e. we declare all permutations to have exactly one subsequence of length 0). We will now find a statistic expressed as a linear combination of ι k 's which is equidistributed with the continued fraction (4.1). We will derive this statistic using the Catalan continued fraction framework of Brändén-Claesson-Steingrímsson [3]. Let A = {A : N × N → Z : A nk = 0 for all but finitely many k for each n} be the ring of infinite matrices with a finite number of non-zero entries in each row. Note in particular that the matrices in A are indexed starting from 0. With each A ∈ A associate a family of statistics { ι, A k } k≥0 where ι = (ι 0 , ι 1 , . . . ), A k is the k th column of A, and Let q = (q 0 , q 1 , . . . ), where q 0 , q 1 , . . . are indeterminates. For each A ∈ A define Note that inc is not a Mahonian statistic.
Proof. Let σ ∈ S n (312) and consider the inflation form σ = 213[σ 1 , 1, σ 2 ] where σ 1 ∈ S k (312) and σ 2 ∈ S n−k−1 (312). Then for each ρ ∈ P we get the recursive relations  Recall the Simion-Schmidt bijection φ : S n (123) → S n (132) which maps σ ∈ S n (123) to the unique permutation in S n (132) with the same left-to-right minima in the same positions as σ (cf Lemma 2.9). As explicitly noted by Claesson and Kitaev [10] this bijection clearly preserves the head statistic and hence [123] head = [132] head . Although head is not a Mahonian statistic we complete its st-Wilf classification below for all subsets of S 3 of size at most three. Equivalences for subsets of larger size can easily be found using similar analysis on the inflation forms. These are less interesting and omitted for brevity. We note in particular that the single pattern distributions with respect to the head statistic are given by well-known refinements of the Catalan numbers.
Proof. The map ψ : S n (321) → S n (312) given by ψ(σ) = φ(σ c ) c , where φ : S n (123) → S n (132) is the Simion-Schmidt bijection, clearly satisfies head(ψ(σ)) = head(σ). Hence [321] head = [312] head . Let σ = a 1 a 2 · · · a n ∈ S n (132). According to the non-recursive description of the standard bijection ∆ : S n (132) → D n (due to Krattenthaler [21]), when a i is read from left to right we adjoin as many U -steps as necessary to the path obtained thus far to reach height h j + 1, followed by a D-step to height h j . Here h j is the number of letters in a j+1 · · · a n which are larger than a j . As such, the number of permutations σ ∈ S n (132) with head(σ) = k is given by the number of Dyck paths starting with exactly n − k + 1 number of U -steps. These are equivalently enumerated by the number of lattice paths with steps (1, 0) and (0, 1) from (1, n − k + 1) to (n, n) staying weakly above the line y = x. By [22,Theorem 10.3.1] the number of such paths are given by n + n − 1 − (n − k + 1) n − (n − k + 1) − n + n − 1 − (n − k + 1) n − 1 + 1 = C n−1,k−1 .
If σ ∈ S n (132, 231), then σ is either decomposed as 12[σ 1 , 1] or as 21[1, σ 1 ] where σ 1 ∈ S n−1 (132, 231). Thus the letters 1, 2, . . . , n are in reverse order recursively placed at the beginning or at the end of the permutation. For σ to have head(σ) = k, the letters k + 1, . . . , n must be placed in increasing order at the end and k at the beginning. Remaining k − 1 letters may be placed on either end giving two choices each (except for the last letter). Hence there exists 2 k−2 permutations σ ∈ S n (132, 231) with head(σ) = k for k > 1.

Summary and conjectures
In Table 2 we summarize the equidistributions proved in this paper (listed in black). In a given cell corresponding to stat row and stat col , a pair of patterns π 1 , π 2 denotes the equidistribution σ∈Sn(π1) q stat row (σ) σ∈Sn(π2) q stat col (σ) .
The equidistributions corresponding to stat row = maj = stat col and stat row = inv = stat col were proved in [13]. The equidistributions between maj, bast ′ and bast ′′ can be proved in a similar way to Proposition 2.1, since the inverse map is the right bijection in two of the cases and the rest can be deduced via the maj-Wilf equivalences from [13]. Remaining equidistributions were either proved directly or follow by combining equidistributions proved in this paper.