On two-sided gamma-positivity for simple permutations

Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis. This conjecture has been proved recently by Lin. We conjecture that an analogous statement holds for simple permutations, and use the substitution decomposition tree of a permutation (by repeated inflation) to show that this would imply the Gessel-Lin result. We provide supporting evidence for this stronger conjecture.


Introduction
Eulerian numbers enumerate permutations according to their descent numbers. The twosided Eulerian numbers, studied by Carlitz, Roselle, and Scoville [6] constitute a natural generalization. These numbers count permutations according to their number of descents as well as the number of descents of the inverse permutation.
See [12, pp. 72, 78] for details. Foata and Schützenberger [7] proved that the coefficients γ n,j are actually non-negative integers. The result of Foata and Schützenberger was reproved combinatorially, using an action of the group Z n 2 on S n which leads to an interpretation of each coefficient γ n,j as the number of orbits of a certain type. This method, called "valley hopping", is described in [8,4]. A nice exposition appears in [11]. Now let A n (s, t) be the two-sided Eulerian polynomial A n (s, t) = π∈Sn s des(π) t ides(π) .
It is well known (see, e.g., [11, p. 167]) that the bivariate polynomial A n (s, t) satisfies as well as A n (s, t) = A n (t, s).
In fact, (1) follows from the bijection from S n onto itself taking a permutation to its reverse, while (2) follows from the bijection taking each permutation to its inverse. A bivariate polynomial satisfying Equations (1) and (2) will be called (bivariate) palindromic of darga n − 1. Note that if we arrange the coefficients of a bivariate palindromic polynomial in a matrix, then this matrix is symmetric with respect to both diagonals. Example 1.1. The two-sided Eulerian polynomial for S 4 is: Its matrix of coefficients is     1 0 0 0 0 10 1 0 0 1 10 0 and is clearly symmetric with respect to both diagonals.
It can be proved (see [12, p. 78]) that the set of bivariate palindromic polynomials of darga n − 1 is a vector space of dimension (n + 1)/2 · (n + 2)/2 , with bivariate gamma basis A bivariate palindromic polynomial is called gamma-positive if all the coefficients in its expression in terms of the bivariate gamma basis are nonnegative. Gessel (see [4,Conjecture 10.2]) conjectured that the two-sided Eulerian polynomial A n (s, t) is gamma-positive. This has recently been proved by Lin [10]. Explicitly: An explicit recurrence for the coefficients γ n,i,j was described by Visontai [15]. This recurrence does not directly imply the positivity of the coefficients, but Lin [10] managed to use it to eventually prove Gessel's conjecture. Unlike the univariate case, no combinatorial proof of Gessel's conjecture is known.
Simple permutations (for their definition see Section 2) serve as building blocks of all permutations. We propose here a strengthening of Gessel's conjecture, for the class of simple permutations.  is gamma-positive, where Simp n is the set of simple permutations of length n.
Using the substitution decomposition tree of a permutation (by repeated inflation), we show how this cojecture implies the Gessel-Lin result. A combinatorial proof of the conjecture will give a combinatorial proof of the Gessel-Lin result. We also provide supporting evidence for this stronger conjecture.
The rest of the paper is organized as follows. Section 2 contains background material concerning simple permutations, inflation, and the substitution decomposition tree of a permutation. In Section 3 we introduce combinatorial involutions on the tree, and use them to give a combinatorial proof of Gessel's conjecture for a certain class of permutations, H(5)∩S n . In Section 4 we show how, more generally, Lin's theorem (Gessel's conjecture) follows combinatorially from Conjecture 1.3. Finally, in Section 5, we give a formula for simp n (s, t) which may have independent value.

Simple permutations and inflation
We start by presenting some preliminaries concerning simple permutations, inflation and the substitution decomposition tree. Original papers will be mentioned occasionally, but terminology and notation will follow (with a few convenient exceptions) the recent survey [14]. Definition 2.1. Let π = a 1 . . . a n ∈ S n . A block (or interval) of π is a nonempty contiguous sequence of entries a i a i+1 . . . a i+k whose values also form a contiguous sequence of integers.
Example 2.2. If π = 2647513 then 6475 is a block but 64751 is not.
Each permutation can be decomposed into singleton blocks, and also forms a single block by itself; these are the trivial blocks of the permutation. All other blocks are called proper.

Definition 2.3. A permutation is simple if it has no proper blocks.
Example 2.4. The permutation 3517246 is simple.
The simple permutations of length n ≤ 2 are 1, 12 and 21. There are no simple permutations of length n = 3. Those of length n = 4 are 2413 and its inverse (which is also its reverse). For length n = 5 they are 24153, 41352, their reverses and their inverses (altogether 6 permutations).
Definition 2.5. A block decomposition of a permutation is a partition of it into disjoint blocks.
For example, the permutation σ = 67183524 can be decomposed as 67 1 8 3524. In this example, the relative order between the blocks forms the permutation 3142, i.e., if we take for each block one of its digits as a representative then the set of representatives is orderisomorphic to 3142. Moreover, the block 67 is order-isomorphic to 12, and the block 3524 is order-isomorphic to 2413. These are instances of the concept of inflation, defined as follows.
Definition 2.6. Let n 1 , . . . , n k be positive integers with n 1 + . . . + n k = n. The inflation of a permutation π ∈ S k by permutations α i ∈ S ni (1 ≤ i ≤ k) is the permutation π[α 1 , . . . , α k ] ∈ S n obtained by replacing the i-th entry of π by a block which is order-isomorphic to the permutation α i on the numbers A very important fact is that inflation is additive on both des and ides. Two special cases of inflation, deserving special attention, are the direct sum and skew sum operations, defined as follows.
Definition 2.11. A permutation is sum-indecomposable (respectively, skew-indecomposable) if it cannot be written as a direct (respectively, skew) sum.
The following proposition shows that every permutation has a canonical representation as an inflation of a simple permutation. . Then there exist a unique integer k ≥ 2, a unique simple permutation π ∈ S k , and a sequence of permutations . . , α k are also unique. If π = 12 (π = 21) then α 1 , α 2 are unique as long as we require, in addition, that α 2 is sum-indecomposable (respectively, skew-indecomposable). Remark 2.14. The additional requirements for π = 12 and π = 21 are needed for uniqueness of the expression. To see that, note that the permutation 123 can be written as 12 [12,1] = 12 3 but also as 12 [1,12] = 1 23. The first expression is the one preferred above (with α 2 sumindecomposable).
One can continue the process of decomposition by inflation for the constituent permutations α i , recursively, until all the resulting permutations have length 1. In the example above, 3412 can be further decomposed as 3412 = 21 [12,12], so that σ = 2413[21 [12,12] [1,1]]. This information can be encoded by a tree, as follows.
Definition 2.15. Represent each permutation σ by a corresponding substitution decomposition tree T σ , recursively, as follows.
• If σ = 1 ∈ S 1 , represent it by a tree with one node.
Example 2.16. Figure 1 depicts the substitution decomposition tree T σ for σ = 452398167. For clarity, the leaves are labeled by the corresponding values of the permutation σ, instead of simply 1.
Inflation can be extended to sets of permutations (an operation called wreath product in [3]).
Definition 2.17. Let A and B be sets of permutations. Define The inflation operation is associative. Defining C 1 = C and C n+1 = C[C n ], we clearly have

Gamma-positivity for H(5)
In this section we present a combinatorial proof of Gessel's conjecture (Lin's theorem) for the subset H(5) ∩ S n of S n (for any positive n).
Fu, Lin and Zeng [9] proved the following (univariate) gamma-positivity result. By Observation 2.8, if π ∈ H(2) then des(π) = ides(π). Hence, one can conclude the following restricted version of Gessel's conjecture for the set of separable permutations. In order to extend Theorem 3.2 further, let us introduce some more definitions.   (1) Each leaf is labeled by 1.
(2) Each internal node is labeled by a simple permutation ( = 1), and the number of its children is equal to the length of the permutation. (3) The labels in each BRC alternate between 12 and 21. Denote by GT n the set of all G-trees with n leaves.
Lemma 3.6. The map f n : S n → GT n sending each permutation σ to its substitution decomposition tree T σ , as in Definition 2.15, is a bijection.
Proof. Follows immediately from Proposition 2.12. The last condition in Definition 3.5 reflects the extra restrictions for the cases π ∈ {12, 21} in Proposition 2.12. Let T = T π be a G-tree, and let {C i | 1 ≤ i ≤ r odd (T )} be the set of all BRC of odd length in T . For each i, let φ i (T ) be the tree obtained from T by switching 12 and 21 in each of the nodes of C i . (A similar action was introduced in [9] for univariate polynomials.) Clearly, each operator φ i is an involution, and the various φ i commute. By Observation 2.8, each φ i changes both des(π) and ides(π) by ±1.
Example 3.7. Consider π = 6713254. The corresponding tree T = T π appears on the left side of Figure 2, and has r odd (T ) = 2. If C 1 is the unique BRC of length 3 in T , then φ 1 (T ) is the tree on the right side of the figure. The permutation corresponding to φ 1 (T ) is 1257634. Note that φ 1 decreased both des(π) and ides(π) by 1.
Let T = T π be a G-tree, and let l 1 , . . . , l k be the labels of nodes in T that belong to the set Simp 4 = {2413, 3142}. Define ψ j (T ) (1 ≤ j ≤ k) to be the tree obtained from T by switching the label l j from 2413 to 3142, or vice versa. Again, it is easy to see that the ψ j are commuting involutions, and each ψ j commutes with each φ i . Switching from 2413 to 3142 increases des(π) by 1 while decreasing ides(π) by 1.
Remark 3.8. Each of the 6 simple permutations π ∈ Simp 5 has des(π) = ides(π) = 2, and we don't need to define involutions for them. Definition 3.9. For any two G-trees T 1 and T 2 , write T 1 ∼ T 2 if T 2 can be obtained from T 1 by a sequence of applications of the involutions φ i and ψ j .
Clearly ∼ is an equivalence relation, partitioning the set GT n (equivalently, the group S n ) into equivalence classes. Definition 3.10. For each equivalence class in GT n , let T 0 be the unique tree in this class in which each odd BRC begins with 12 and each node representing a simple permutation of length 4 is labeled 2413. The corresponding permutation π 0 has the minimal number of descents in its class. The tree T 0 and the permutation π 0 are called the minimal representatives of their equivalence class. Proof. Let π 0 be the minimal representative of A, and let T 0 = T π0 . For i ∈ {2, 4, 5}, let v i be the number of nodes of T 0 having labels of length i. Since T 0 has exactly n leaves, its total number of nodes (including leaves) is On the other hand, counting the children of each node gives v − 1 = i∈{2,4,5} iv i .

It follows that
n − 1 = v 2 + 3v 4 + 4v 5 . Let r = r odd (T 0 ) be the number of odd BRC in T 0 , and let d 2 be the number of nodes labeled 21. By definition, each BRC alternates between 12 and 21 and each odd BRC in T 0 starts with 12. It follows that v 2 = 2d 2 + r, so that n − 1 = r + 2d 2 + 3v 4 + 4v 5 .
Define a BRC of T (in analogy to Definition 3.3) to be a maximal nonempty chain of consecutive right descendants, all labeled 2. How can we recover a permutation σ ∈ A(T ) from the tree T ? Each internal node, labeled by a number , can be relabeled by any simple permutation of length , with the single restriction that the labels in each BRC must alternate between 12 and 21, starting with either of them. It thus follows, by Observation 2.8, that for each simplified tree T , the polynomial is a product of factors, as follows: • Each internal node with label ≥ 4 contributes a factor simp (s, t).
• Each BRC of even length 2k contributes a factor 2(st) k .
• Each BRC of odd length 2k + 1 contributes a factor (st) k (1 + st). By Conjecture 1.3, all those factors are gamma-positive, and so is their product. Summing over all equivalence classes in S n completes the proof.
It is clear from the arguments above that a combinatorial proof of Conjecture 1.3 will immediately yield a combinatorial proof of Theorem 1.2. In fact, the preceding section contains such a combinatorial proof assuming there are only labels ≤ 5, using simp 4 (s, t) = st(s + t) and simp 5 (s, t) = 6(st) 2 . We were unable to extend the combinatorial arguments to length 6, although the corresponding polynomial is indeed gamma-positive: In fact, Conjecture 1.3 has been verified by computer for all n ≤ 12.

The bi-Eulerian polynomial for simple permutations
In [2], the ordinary generating function for the number of simple permutations was shown to be very close to the functional inverse of the corresponding generating function for all permutations. In this section we refine this result by considering also the parameters des and ides, thus obtaining a formula for simp n (s, t).
Recall from Definition 2.11 the notions of sum-indecomposable and skew-indecomposable permutations. Note that the summation in the definition of S(x, s, t) is only over n ≥ 4. We want to find relations between these generating functions.
From now on, we consider F (x, s, t) etc. as formal power series in x, with coefficients in the field of rational functions Q(s, t). We therefore use the short notation F (x), or even F . For example, the composition S • F means that F is substituted as the x variable of S(x, s, t). By Proposition 2.12 and Observation 2.8, Note that the reversal map π → π , defined by π (i) = n − 1 − π(i) (1 ≤ i ≤ n), is a bijection from S n onto itself (and also from I + n onto I − n ), satisfying des(π ) = n − 1 − des(π) and ides(π ) = n − 1 − ides(π). Therefore: we finally obtain a formula for simp n (s, t).