Inversion Formulae on Permutations Avoiding 321

We will study the inversion statistic of 321-avoiding permutations, and obtain that the number of 321-avoiding permutations on [n] with m inversions is given by |S n,m (321)| = b⊢m n − ∆(b) 2 l(b). where the sum runs over all compositions b = (b l(b) = k is the length of b, and ∆(b) := |b 1 | + |b 2 − b 1 | + · · · + |b k − b k−1 | + |b k |. We obtain a new bijection from 321-avoiding permutations to Dyck paths which establishes a relation on inversion number of 321-avoiding permutations and valley height of Dyck paths.

In this paper, we will study the inversion distribution of 321-avoiding permutations.
Motivated by [2,6], in this paper we will study the inversion distribution of 321avoiding permutations.As the main result, we give an explicit formula counting the number of 321-avoiding permutations with the fixed inversion number.We also find a bijection between 321-avoiding permutations and Dyck paths, which is new to the best of our knowledge.From this bijection, we show that the inversion number of 321-avoiding permutations and the valley-sum of Dyck paths are equally distributed.

Inversions of Permutations Avoiding 321
For 1 k n, let I k n (321, q) be the generating function defined by q inv(σ) .
Then we have I n (321, q) = I 1 n (321, q) and I n n (321, q) = 1 for all n 1.
Lemma 1.For 1 k n, we have n (321).So this case contributes a term I k+1 n (321, q) to the generating function Changing the index i to k − i, the proof will be complete by combining (i) and (ii).
In order to characterize the generating function I n (321, q) as a counting function of lattice points in a lattice polytope, we introduce the following lemma.Lemma 2. Assuming x 0 = 0, for all 1 t n, we have Proof.The statement is true for t = 1 by Lemma 1.To use induction on t, suppose the above equality holds for t.From Lemma 1, we have Using above formula to substitute the term I t+1−xt n+1−xt (321, q) in the formula of this Lemma, we can easily conclude that the equality holds for the case t + 1.
Let Ω n be a convex lattice polytope defined by Recall that I n+1 (321, q) = I 1 n+1 (321, q) and I n+1−xn n+1−xn (321, q) = 1.From above lemma by taking t = n we can easily obtain Proposition 3. Assuming x 0 = 0, we have In the following we will give a more explicit interpretation about this formula.Let inv k (σ) be the number of inversions of σ whose first element is k, i.e, From the definition of I n+1 (321, q) and Proposition 3, we have Below we recursively define a map the electronic journal of combinatorics 22(4) (2015), #P4.28 such that x 1 = inv 2 (σ) and for k 2, Figure .1 shows an example, where the second vector is inv 1 (σ), . . ., inv 9 (σ) .
Indeed, for x = (x 1 , . . ., x n ) ∈ Ω n , assuming x 0 = 0 and x n+1 = n + 1, we construct a Dyck path D x as follows.By reading i from 1 to n + 1, for each i we add an up-step and x i − x i−1 down-steps from left to right.Combining with Theorem 4, we can easily obtain our first main result.
As an application of Theorem 5, we will give a counting formula on the number of 321-avoiding permutations with a fixed inversion number.For any D ∈ D n , we define a the electronic journal of combinatorics 22(4) (2015), #P4.28 tunnel of D to be a horizontal segment between two lattice points of D that intersects D only in these two points, and stays always below D. From Theorem 5, for m 0, there is a bijection . Let l i be the length of the path D located between the i-th and (i + 1)-th valley, for i = 0, 1, 2, • • • , k.Then we have Where t i is the number of tunnels between the i-th and (i + 1)-th valley.Let a 0 = 0 and a k+1 = 0 be the heights of the starting point and the terminal point of the Dyck path D, respectively.Write S k n (321) be the collection of 321-avoiding permutations of [n] and containing 12 • • • k as a subsequence,

Figure. 2
presents an example.It is obvious that this construction gives an bijection from Ω n+1 to D n+1 .If all valleys of a Dyck path D have heights a 1 , . . ., a k , denote by v(D) = k i=1 (a i + 1).