As for Theorem 4.1.2 and Corollary 4.3.1, our proofs make use of recurrence relations of multivariate generating functions. So far we do not have a combinatorial proof. As shown in [18], there are a number of combinatorial structures bijective with Baxter permutations. We are interested in any connection leading to a combinatorial proof of these results.
Speaking of the phenomenon that the signed enumerator of objects of size 2n is es-sentially equal to the ordinary enumerator of objects of size n, one may ask if the phe-nomenon holds for other classes of permutations, say Sn, or other statistics. However, this phenomenon seems rare. Note that Bn(ω) = Sn(ω) for ω ∈ {132, 213, 231, 312}, and there is no such identities for the maj-signed des-enumerators for these Bn(ω), or Bn itself.
Another natural question is that if we restrict the Carlitz’s identity to B2n+1(321) is there still a closed form for the generating function
P
π∈B2n+1(321)tmaj(π)qdes(π) Qn
i=0(1 − qti) .
As for the proof of Theorem 4.1.3, for the purpose of providing recursive bijection with Rn, the generating tree for Wn presented in Proposition 4.4.1 is somewhat
artificially constructed, with complicated pattern-combinations for nodes and word-replacements for successors. In fact, the generating tree for a class of combinatorial objects is not unique. As pointed out by a referee, another choice of the generating tree for Wn is
ω = ε (root) (30) (21)(31)(32) ω = σ1σ2∗ (1) (21)
ω = σ1, σ1σ1∗, σ1σ3∗ (21) (21)(31) ω = σ2σ3∗ (22) (1)(31) ω = σ2, σ2σ1∗, σ2σ2∗ (31) (1)(31)(32) ω = σ3∗ (32) (21)(22)(32),
where the stable words ω are classified by their prefixes shown in the first column.
Hence there must be an isomorphic tree for Rn by a certain classification that follows these succession rules while providing the requested generating function. We leave it to interested readers.
One may also ask if the connection between 321-avoiding Baxter permutations and the positive braid words on four strands can be extended further. To our knowledge, it is so far an isolated property and we do not have any generalization.
Chapter 5 Involutions
5.1 Introduction
The study of sign-balance of combinatorial objects is of frequent interest in combi-natorics. Some enumerative results reveal an interesting phenomenon (up to small variations) that the signed enumerator of objects of size 2n is essentially equal to the ordinary enumerator of objects of size n. This phenomenon can be expressed as
X
π∈X2n or X2n+1
(−1)stat1(π)qstat2(π)= f (q) X
π∈Xn
q2·stat2(π),
where Xn is a family of objects of size n with statistics stat1 and stat2, and f (q) is a rational function. When dealing with permutations, stat1 is usually a Mahonian statistic. In this paper, we present two instances of such a phenomenon on 321-avoiding involutions.
5.1.1 Enumeration of 321-avoiding involutions
Let Sn be the set of permutations of [n] := {1, 2, . . . , n}, and let Sn(321) ⊆ Sn be the set of permutations without decreasing subsequence of length three, called 321-avoiding. Recall that a permutation σ ∈ Sn is called an involution if and only if σ−1 = σ, and equivalently the cycle structure of σ contains no cycle of length greater than two. Let In(321) be the set of involutions in Sn(321). Simion and Schmidt [29]
completed the enumeration of involutions avoiding a pattern of length three, including
|In(321)| =
n
n 2
, the nth central binomial coefficient.
For a permutation σ = σ1· · · σn ∈ Sn, let fix(σ) be the number of fixed points of σ, i.e., fix(σ) = |{i : σi = i, 1 ≤ i ≤ n}|, and let exc(σ) be the number of excedances, i.e., exc(σ) = |{i : σi > i, 1 ≤ i ≤ n}|. The descent set of σ is defined as Des(σ) = {i : σi > σi+1, 1 ≤ i ≤ n − 1}, and the descent number (des) and major
index (maj) of σ are defined by des(σ) = |Des(σ)| and maj(σ) =P
i∈Des(σ)i. Recently, Barnabei et al [4] proved that the distribution of the major index over 321-avoiding involutions is given by
5.1.2 A quick review on refined sign-balance
Simion and Schmidt [?] initiated the study of sign-balance for pattern-avoiding per-mutations, who proved that the number of even and odd permutations in Sn(321) are equal if n is even, and differ (up to a sign) by a Catalan number otherwise.
D´esarm´enien and Foata [13] and Wachs [35] investigated refined sign-balance on Sn, respecting the number of descents. Simion and Gessel studied the refined sign-balance on Sn, respecting the major index (see [35, Corollary 2]). Robertson, Saracino and Zeilberger [27] started the refined enumeration of pattern-avoiding permutations, re-specting the number of fixed points and excedances. These results sparked further work on refined signed enumeration for restricted permutations.
Adin and Roichman [2] obtained the following refined sign-balance identities for Sn(321), respecting the statistic ldes(σ), the last descent, of σ ∈ Sn(321), i.e., ldes(σ) = max{i : σi > σi+1, 1 ≤ i ≤ n − 1}. the same time, Mansour [23] found variation of the identities on 132-avoiding permu-tations, and shortly afterward, Reifegerste [26] proved analogous identities combina-torially respecting the length of the longest increasing subsequence of π ∈ Sn(321).
Recently, the authors obtained an analogous result for 321-avoiding alternating per-mutations, respecting the first (resp. last) entry of the permutations [16], and ob-tained a bivariate refinement of the major-balance enumerator for 321-avoiding Bax-ter permutations [17]. The aim of this paper is to study the refined major-balance enumerator for 321-avoiding involutions.
5.1.3 Main results
The initial terms of the bivariate maj-fix enumerator and maj-des enumerator for In(321) are
One of the main results in this paper is the following major-balance identities for In(321).
Note that (i) and (ii) of Theorem 5.1.1 are essentially equivalent since fix(σ) + 2 · exc(σ) = n for any σ ∈ In(321). The other main result is the following.
We shall prove Theorems 5.1.1 and 5.1.2 by a combinatorial approach. A lattice path from (0, 0) to (n, n) in the plane Z × Z, using north step N = (0, 1) and east step E= (1, 0), that never goes below the line y = x is called a Dyck path of length 2n. A partial Dyck path is a lattice path not going below the line y = x. We make use of a bijection between 321-avoiding involutions and partial Dyck paths, given by Deutsch, Robertson and Saracino in [14], as the initial stage of this approach.
To prove Theorem 5.1.1, we transform a partial Dyck path into an ordered tree whose non-root internal nodes have exactly two children, called cluster 2-tree. We find a statistic ℓ(T ) of a cluster 2-tree T that corresponds to the major index of 321-avoiding involutions. Then we prove Theorem 5.1.1 by establishing an ℓ-parity-reversing involution on cluster 2-trees (see Proposition 5.3.7).
To prove Theorem 5.1.2, we make use of a bijection given by Barnabei et al [4]
to transform a partial Dyck path of length n into a lattice path going from (0, 0) to (⌊n2⌋, ⌈n2⌉) using N and E steps without restriction, called a grand Dyck path of length n. It is known that the area above the grand Dyck path within the rectangle coincides with the major index of 321-avoiding involutions. Then we prove Theorem 5.1.2 by establishing an area-parity-reversing involution on grand Dyck paths (see Propositions 5.5.4, 5.5.8 and 5.5.11).
The rest of this paper is organized as follows. In section 2 we establish a two-stage bijection that links 321-avoiding involutions to cluster 2-trees. In section 3 we establish an ℓ-parity-reversing involutions on cluster 2-trees, which leads to a proof of Theorem 5.1.1. In section 4 we describe a bijection that links partial Dyck paths to grand Dyck paths given in [4]. In section 5 we establish an area-parity-reversing involution on grand Dyck paths, which leads to a proof of Theorem 5.1.2.