Adin and Rochiman noticed the phenomenon that the signed enumerator of objects of size 2n is essentially equal to the ordinary enumerator of objects of size n, and
they proved a refined sign-balance property on Sn(321) respecting the statistic ldes.
They also studied the statistic ‘the position of the entry n in σ ∈ Sn(321)’, denoted by lind, and proved the equidistribution of ldes and lind on the set Sn(321). (See also [25] for Reifegerste’s bijective proof.) We encounter the same phenomenon when we consider the sign-balance of 321-avoiding alternating permutations. As a variant of Adin and Rochiman’s result, we prove refined sign-balance properties on Altn(321) respecting the statistics lead and end. It is known that the statistics lead and end on the set Alt2n(321) are essentially equidistributed with the statistic ldes on the set Sn(321). However, these statistics are quite different in nature. Translated in terms of Dyck paths, these statistics have the following distinct interpretations. (We refer the readers to [2, Section 5] for a bijection between Sn(321) and Cn that use standard Young tableaux as an intermediate stage.) If P = x1x2· · · x2n ∈ Cn is the Dyck path that corresponds to σ ∈ Sn(321), then it is shown that
ldes(σ) = max{i : xi = N, xi+1 = E, 1 ≤ i ≤ n − 1}, (where ldes(σ) = 0 for P = N · · · NE · · · E) lind(σ) = max{i : xi = N, xi+1 = E, 1 ≤ i ≤ n}.
If P corresponds to σ ∈ Alt2n(321) under the bijection Ω : Alt2n(321) → Cn in Proposition 3.2.1, then we have
lead(σ) = min{i : xi−1 = N, xi = E, 2 ≤ i ≤ n + 1}, end(σ) = max{i : xi = N, xi+1 = E, n ≤ i ≤ 2n − 1}.
Taking into account the sign of permutations, we are interested to know if there are explicit connections among these statistics that show the equivalence of Theorem 4.1.1 and the assertions (i) and (iii) of Theorem 3.1.2 (or Theorem 3.1.3). We leave the question to the readers.
Chapter 4
Baxter permutations
4.1 Introduction
4.1.1 Baxter permutations and Pattern avoidance.
Let Sn be the set of all permutations of {1, · · · , n}. A permutation π ∈ Sn is called a Baxter permutation if it satisfies the following conditions for all 1 ≤ a < b < c <
d ≤ n,
• if πa+ 1 = πd and πb > πd then πc > πd;
• if πd+ 1 = πa and πc > πa then πb > πa.
For example, 25314 is a Baxter permutation, but 5327146 is not. It is clear from the definition that the reverse of a Baxter permutation is also Baxter. Let Bn be the set of Baxter permutations in Sn. Chung, Graham, Hoggatt, and Kleiman [8] first proved analytically that
|Bn| = 2 n(n + 1)2
Xn k=1
n + 1 k − 1
n + 1 k
n + 1 k + 1
.
A bijective proof is given by Viennot [33]. Other proofs are given by Mallows [21], Dulucq and Guibert [12]. Recently, Felsner, Fusy, Noy, and Orden [18] established explicit bijections between a number of objects that are enumerated by Baxter num-bers.
For permutations π = π1· · · πn∈ Sn and ω = ω1· · · ωt∈ St, t ≤ n, we say that π contains an ω-pattern if there are indices i1 < · · · < it such that πij < πik if and only if ωj < ωk. Moreover, π is called ω-avoiding if π contains no ω-patterns. Let Sn(ω) denote the set of ω-avoiding permutations in Sn. It is known that |Sn(ω)| = Cn =
1 n+1
2n n
, the nth Catalan number, for every ω ∈ S3.
Let Bn(ω) be the set of ω-avoiding permutations in Bn. Mansour and Vajnovszki [24] enumerated the permutations in Bn(123) that avoid (or contain) another pattern
of a certain length. They proved that the generating function for |Bn(123)| is given deter-mined the sign-balance of the set Sn(321)
X π. Making use of a multivariate generating function, Adin and Roichman [2] proved a refinement of Eq. (4.1.1) with respect to the statistic ldes(π), the last descent, of π ∈ Sn(321), i.e., ldes(π) = max{i : πi > πi+1, 1 ≤ i ≤ n − 1}.
Theorem 4.1.1 (Adin-Roichman). The following identities hold.
X
At the same time, Mansour [23] found variation of the identities for Sn(132), and shortly afterward, Reifegerste [26] proved analogous identities combinatorially re-specting the length of the longest increasing subsequence of π ∈ Sn(321). Recently, the authors obtained an analogous result for 321-avoiding alternating permutations, respecting the first (resp. last) entry of the permutations [16].
These results reveal a phenomenon that the signed enumerator of objects of size 2n is essentially equal to the ordinary enumerator of objects of size n. One would nat-urally ask if there is any similar identity on other classes of (restricted) permutations, but computer checks show that this phenomenon seldom appears.
In the first part of this paper we will present a variant of refined balance identities for 321-avoiding Baxter permutations, where the sign of a permutation depends on the parity of its major index. To wit, we are considering the refined ‘maj-balance’ of Bn(321). Note that the notion of maj-balance was also mentioned in [31].
We consider the following statistics of π = π1· · · πn ∈ Sn. Let fix(π) be the number of fixed points of π, i.e., fix(π) = |{i : πi = i, 1 ≤ i ≤ n}|. The descent set of
π is defined as Des(π) = {i : πi > πi+1, 1 ≤ i ≤ n − 1}, and the descent number (des) and major index (maj) of π are defined by des(π) = |Des(π)| and maj(π) =P
i∈Des(π)i.
The statistics maj and des play an important role in combinatorics. For example, the celebrated Carlitz identity [7]
has stimulated a lot of work.
Our first main result is the following maj-fix-des version of refined maj-balance identity for Bn(321).
We will prove this result via recurrence relation of multivariate generating func-tions. For the case of even length, we show that the maj-signed fix-des-enumerator of B2n(321) can be expressed as a combination of the ordinary fix-des-enumerators of Bi(321) for all i = 0, 1, . . . , n. (See Theorem 4.2.5.) Moreover, we will also obtain a counterpart result on B2n+1(123). (See Corollary 4.3.1)
4.1.3 Positive braid words.
The Braid group on n strands is the group generated by {σ1, · · · , σn−1} with the
relations (
σiσj = σjσi for |i − j| ≥ 2;
σiσjσi = σjσiσj for |i − j| = 1. (4.1.3) Note that if we add the relation σi2 = e, then we get a presentation of the symmetric group Sn on n elements. A braid word is a finite sequence of letters σi and their inverses σi−1. A positive braid word is a braid word that contains no letter σi−1. The set of positive braid words is also known as the monoid of positive braids, with concatenation as multiplication. For positive braid words, the length is an invariant, which is equal to the number of their letters. We refer the readers to [20] for more information.
To our surprise, by a search in OEIS [30], the generating function for |Bn(123)| =
|Bn(321)| is very close to the one for the number of positive braid words on four strands (i.e., with letters {σ1, σ2, σ3}). It has appeared in [30, A097550] that the generating function for the number of such words of length n is given by
1 + z2
1 − 3z + 2z2− z3 = 1 + 3z + 8z2+ 19z3+ 44z4+ 102z5+ 237z6 + · · · .
Note that this rational function has exactly the same denominator as the generating function for |Bn(123)| = |Bn(321)|.
Back to 321-avoiding Baxter permutations, our second main result is to establish a connection between the 321-avoiding Baxter permutations with the entry 1 preceding the entry 2 and the positive braid words on four strands.
Theorem 4.1.3. There is a bijection between the permutations π ∈ Bn+2(321) with the condition π−1(1) < π−1(2) and the positive braid words on four strands of length n.
To prove this theorem, we will construct generating trees for both families and show that they share the same succession rules. Readers are referred to [9, 34] for more information on generating trees.
The rest of the paper is organized as follows. In Section 2 we will first deal with B2n+1(321) and prove Theorem 4.1.2, then derive a similar identity on B2n(321). In Section 3 we will obtain a counterpart result on B2n+1(123). In Section 4 we will prove Theorem 4.1.3.