A new encoding of permutations by Laguerre histories

We construct a bijection from permutations to some weighted Motzkin paths known as Laguerre histories. As one application of our bijection, a neat $q$-$\gamma$-positivity expansion of the $(\inv,\exc)$-$q$-Eulerian polynomials is obtained.


Introduction
A Motzkin path of length n is a lattice path in the first quadrant starting from (0, 0), ending at (n, 0), with three possible steps: U = (1, 1) (up step), L = (1, 0) (level step) and D = (1, −1) (down step). A 2-Motzkin path is a Motzkin path in which each level step is labelled by L 0 or L 1 . The 2-Motzkin paths will be represented as words over the alphabet {U, D, L 0 , L 1 }. A Laguerre history of length n is a pair (w, µ) such that w = w 1 · · · w n is a 2-Motzkin path and µ = (µ 1 , · · · , µ n ) is a vector satisfying 0 ≤ µ i ≤ h i (w), where h i (w) := #{j | j < i, w j = U} − #{j | j < i, w j = D} is the height of the i-th step of w. Denote by L n the set of all Laguerre histories of length n. It is known that the cardinality of L n is (n + 1)!.
Laguerre histories can be used to encode permutations. Two famous such encodings in the literature are known as the Françon-Viennot bijection and the Foata-Zeilberger bijection; see [3] for the relationship between these two bijections and [3,8,9] for other modifications of them. The purpose of this paper is to present a new encoding of permutations by Laguerre histories with an interesting application to the q-γ-positivity expansion of the (inv, exc)-q-Eulerian polynomials. The inspiration of our bijection comes from the recent works by Cheng-Elizalde-Kasraoui-Sagan [2], Lin [5] and Lin-Fu [6]. We need some further definitions and notations before we can state our main results.
Theorem 1 (Shin and Zeng). For n ≥ 1, we have For a 2-Motzkin path w = w 1 · · · w n of length n, define Let area(w) be the area between w and the x-axis. Our new encoding of permutations by Laguerre histories is a generalization of the bijection in [6,Lemma 16] between 321-avoiding permutations and 2-Motzkin paths.
Theorem 2. There is a bijection Φ : S n → L n−1 such that if Φ(σ) = (w, µ), then where EXC(σ) is the set of excedances of σ. As one application of our encoding Φ, the following neat q-γ-positivity expansion, different with that in Theorem 1, for the (inv, exc)-q-Eulerian polynomials is derived.
As an example of Theorem 3, for n = 4, it follows from (1.4) that We will also provide an alternative approach to Theorem 1 by combining our bijection Φ and a modified version of the Françon-Viennot bijection. Denote by A n the set of permutations σ ∈ S n that is down-up: It is well known (cf. [12]) that |A 2k−1 | is the k-th tangent number T k , which appears in the Taylor expansion Setting t = −1 in Theorem 1 we recover the following result about q-tangent numbers due to Shin and Zeng [8,Theorem 3].
Corollary 4 (Shin and Zeng). For n ≥ 1, we have In the same vein, setting t = 1 in Theorem 3 gives the following new interpretation of the above q-tangent numbers.
if n is even, Although combining our bijection Φ and a modification of the Françon-Viennot bijection Ψ (introduced in Section 3) will set up a link between DE 2k+1,k and A 2k+1 , no direct bijection between these two models is known.
The rest of this paper is organized as follows. In Section 2, we construct the bijection Φ and prove Theorem 2. In Section 3, we introduce a simple group action on Laguerre histories and prove Theorems 1 and 3.

The construction of Φ
In this section, we will construct the bijection Φ and prove Theorem 2. The following definition is important in constructing the bijection Φ.
We are going to prove that Φ is a bijection between S n and L n−1 satisfying (1.3). The following lemma plays an essential role in proving Φ is a bijection.
It then follows that First we prove the "only if" side. We distinguish two cases: • If p k = 0, that is σ(k) ≤ k, then d k (v, p) counts the number of indices ℓ such that ℓ ≤ k and σ −1 (ℓ) ≥ k. Thus, d k (v, p) ≥ nest k (σ) + 1 according to the definition of nest k (σ). • Otherwise, we have p k = 1 and in this cased k (v, p) counts the number of indices ℓ such that ℓ ≤ k and σ(ℓ) > k. It follows from the definition of nest k (σ) that d k (v, p) ≥ nest k (σ) + 1. In view of (2.2), we get d k (v, p) ≥ nest k (σ) + 1 in either case.
In step (b) of the above algorithm, since µ k + 1 ≤ d k (v, p) =d k (v, p) (in view of (2.2)), j must exist and j > k. In order to have nest k (σ) = µ k the value of σ(k) must be j. Continuing with the above running example, we determine σ(6) = 9, σ(5) = 8, σ(2) = 3 and σ(1) = 4, successively. Finally, the permutation σ constructed by the above two algorithms is 432189765, which coincides with the one in Example 8. The proof of the "if" side is complete. Now, we are ready to prove Theorem 2.
Proof of Theorem 2. It follows from the construction of Φ that Laguerre histories of length n − 1 are in bijection with the triples (v, p, µ), where v = (v 1 , v 2 , . . . , v n ) and p = (p 1 , p 2 , . . . , p n ) are two 0-1 vectors with the same number of zeros , and µ = (µ 1 , µ 2 , . . . , µ n ) is a vector satisfying (2.1). The latter objects are in bijection with S n by Lemma 9 and so Φ is a bijection between S n and L n−1 .
Next we show that Φ has the required properties (1.3). The first equality of (1.3) is clear from the definition of Φ. For the second equality of (1.3), we claim that (2.3) h i (w) = cros i (σ) + nest i (σ).
• Case 1: w i = D or L 0 . In this case, we have p i = 0 and • Case 2: w i = U or L 1 . In this case, we have p i = 1 and Hence we have deduced (2.3), which ends the proof of Theorem 2.

Applications of Φ
Lin [5] introduced a group action on 2-Motzkin paths in the sprit of the Foata-Strehl action on permutations. Here we generalize it to Laguerre histories. Let i ∈ [n] and (w, µ) ∈ L n . If the i-th step of w is level, then let ϕ i (w, µ) = (w ′ , µ), where w ′ is the 2-Motzkin path obtained from w by changing the label of the i-th step. Otherwise, define ϕ i (w, µ) = (w, µ). For any subset S ⊆ [n] define the function φ S : L n → L n by ϕ S (w, µ) = i∈S ϕ i (w, µ). Hence the group Z n 2 acts on L n via the function ϕ S . This action divides the set L n into disjoint orbits and each orbit has a unique Laguerre history which has all its level steps labelled by L 0 . Let us introduce O n,k = {(w, µ) ∈ L n : all level steps of w are L 0 , #U(w) = k}.
It then follows from this action that We are now in position to prove Theorem 3.
Proof of Theorem 3. It is clear from the construction of Φ that a permutation σ ∈ S n has no shifted double excedances if and only if w has no L 1 step, where (w, µ) = Φ(σ). Theorem 3 then follows from Theorem 2 and expansion (3.1).