Rational Dyck Paths in the Non Relatively Prime Case

We study the relationship between rational slope Dyck paths and invariant subsets of $\mathbb Z,$ extending the work of the first two authors in the relatively prime case. We also find a bijection between $(dn,dm)$--Dyck paths and $d$-tuples of $(n,m)$-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and the geometry of affine Springer fibers and knot invariants in the non relatively prime case.


Introduction
Catalan numbers, in one of their incarnations, count the number of Dyck paths, that is, the lattice paths in a square which never cross the diagonal. In recent years, a number of interesting results and conjectures [3,5,6,7,8,9,17] about "rational Catalan combinatorics" have been formulated. An (n, m)-Dyck path is a lattice path in the n × m rectangle which never crosses the diagonal. We will denote the set of all (n, m)-Dyck paths by Y n,m . For coprime m and n there are a number of interesting maps involving Y n,m : (a) J. Anderson constructed a bijection A between Y n,m and the set Core n,m of simultaneous (n, m)core partitions.
(b) Armstrong, Loehr and Warrington defined a "sweep" map ζ : Y n,m → Y n,m and conjectured that it is bijective.
(c) The first two authors defined two maps D and G between Y n,m and the set M n,m of (+n, +m)invariant subsets of Z ≥0 . If combined with a natural bijection between Core n,m and M n,m , the map D coincides with A. Furthermore, one can prove that ζ = G • D −1 .
Remark 0.1 Recently, after the original version of this manuscript was submitted, Nathan Williams posted a preprint [16] containing a proof of the bijectivity of the sweep map. Statements below involving ζ should be modified appropriately.
We illustrate all these maps in Figure 1: bidirected arrows indicate bijections, and directed arrows indicate maps for which the inverse is not known in general. The goal of the present paper is a partial generalization of Figure 1 to the non-coprime case. Let (n, m) be relatively prime, and d be a positive integer. Let N = dn and M = dn. The set Y N,M is well defined for all n, m, d, and the definition of ζ can be carried over with minimal changes. However, the sets Core N,M and M N,M become infinite. Indeed, an (+N, +M )-invariant subset of Z ≥0 can be identified with a collection of d (+n, +m)-invariant subsets in each remainder mod d. These subsets can be arbitrarily shifted with respect to each other. More abstractly, this defines a map π : M N,M → (M n,m ) d and different shifts correspond to different preimages under π.
To resolve this problem, we introduce a certain equivalence relation ∼ on M N,M . If ∆ 1 ∼ ∆ 2 then π(∆ 1 ) = π(∆ 2 ), so π is well defined on M N,M /∼. The following theorem is the main result of the paper.
can be described as following: the N + M steps in an (N, M )-Dyck path are colored in d colors, i.e. by Z/dZ, such that there are n + m steps of the same color i, and these steps will form an (n, m)-Dyck path after possibly shifting by integer multiples of − −−−− → (m, −n) to make them connected. We illustrate all these maps in Figure 2. We also give an explicit description of the "coloring map" col d . In the "classical" case M = N we get d = N and m = n = 1, therefore col d colors a Dyck path in n colors such that the pairs of steps of the same color form an (1, 1) Dyck path. We prove the following: Corollary 0.3 Present a Dyck path as a regular sequence of parentheses. Then every opening and its corresponding closing parenthesis have the same color under col d .

Relatively prime case
Let (n, m) be a pair of relatively prime positive integers. Consider an n × m rectangle R n,m . Let Y n,m be the set of Young diagrams that fit under the diagonal in R n,m . We will often abuse notation by identifying a diagram D ∈ Y n,m with its boundary path (sometimes also called a rational Dyck path), and with the corresponding partition. We will also think about the rectangle R n,m as a set of boxes, identified with a subset in Z ≥0 with the bottom-left corner box identified with (0, 0). In our convention, the boundary path of D ⊂ R n,m follows the boundary from top-left corner (0, n) to the bottom-right corner (m, 0). We will refer to the direction along the boundary path from (0, n) toward (m, 0) as from higher to lower. See Example 1.11 below.
There are two important combinatorial statistics on the set Y n,m : area and dinv .
Then area(D) is equal to the number of whole boxes that fit between the diagonal of R n,m and the boundary path of D.
Note that area(D) ranges from 0 for the full diagram to δ = (m−1)(n−1) 2 for the empty diagram. The co-area(D) = δ − area(D) is then just the number of boxes in the Young diagram D. One natural approach to the dinv statistic is to define the map ζ : Y n,m → Y n,m and then set dinv(D) := area(ζ(D)). In the case m = n + 1 the map ζ was first defined by Haglund ([10]), then it was generalized by Loehr to the case m = kn + 1 for any k ∈ Z ≥0 ( [11]), and to the general case of any relatively prime (n, m) by Gorsky and Mazin in [7]. In [3] it was put into even larger framework of so called sweep maps. Below is one of the possible equivalent definitions. Definition 1.2 The rank of a box (x, y) ∈ Z 2 is given by the linear function Note that the boxes of non-negative ranks are exactly those that fit under the main diagonal of R n,m . Let D ∈ Y n,m . One ranks the steps of the boundary path of D as follows. In other words, the ranks of steps can be defined inductively as follows. We follow the boundary path of D starting from the top-left corner. The first step is ranked 0. Otherwise, the rank of every horizontal step equals the rank of the previous step minus n, and the rank of every vertical step equals the rank of the previous step plus m.
Note that for relatively prime (n, m) all the ranks of the steps of a diagram D ∈ Y n,m are distinct. The definition of the map ζ is illustrated in Example 1.11. One can verify that the diagram ζ(D) fits under the diagonal of R n,m (see [7] and [3]). It was conjectured that the map ζ is a bijection. See Remark 0.1. Some partial progress was made by Haglund, Loehr, Gorsky, and Mazin: Bijectivity of ζ in the general case has attracted a lot of attention: see [17] and [6]. Most recently, Nathan Williams proved bijectivity of ζ for general coprime n, m ( [16]). The following approach to studying ζ was suggested in [7]. Definition 1.6 Let M n,m be the set of subsets ∆ ⊂ Z ≥0 such that ∆ + m ⊂ ∆, ∆ + n ⊂ ∆, and min(∆) = 0. We say ∆ ∈ M n,m is (+n, +m)-invariant.
In [7] two maps D and G from the set M n,m to Y n,m were constructed. Let ∆ ∈ M n,m . Definition 1.7 The diagram D(∆) consists of all boxes that fit under the diagonal in R n,m , and whose ranks belong to ∆.
In particular, one gets that D(Γ n,m ) = ∅, where Γ n,m := {an + bm | a, b ∈ Z ≥0 } is the semigroup generated by m and n, and D(Z ≥0 ) is the full diagram containing all the boxes below the diagonal. Note that the (+n, +m)-invariance of ∆ implies that D(∆) is indeed a Young diagram. Note also that D is a bijection. Indeed, it is not hard to see that rank provides a bijection between the boxes below the diagonal in R n,m and the integers in Z ≥0 \ Γ n,m . Definition 1.8 The numbers 0 = a 0 < a 1 < . . . < a n−1 , such that {a 0 , . . . , a n−1 } = ∆ \ (∆ + n) are called the n-generators of ∆. The numbers Definition 1.10 The diagram G(∆) has row lengths g 0 , . . . , g n−1 given by the following formula: Equivalently, the boundary path of G(∆) is obtained by arranging the set {a 0 , . . . , a n−1 , b 0 , . . . , b m−1 } in decreasing order and replacing n-generators by vertical steps and m-cogenerators by horizontal steps.
It is not hard to conclude from the above definitions that  Figure 3 for the diagrams D and ζ(D). Note, that if one takes the union of the 5-generators and 3cogenerators and reads them in decreasing order, then one gets 9, 7, 6, 4, 3, 2, 0, −3. Replacing generators by "v" and cogenerators by "h", one gets vvvhvhvh, which is the boundary path of ζ(D).
The approach with invariant subsets allows one to relate the dinv statistic to certain varieties that appear in algebraic geometry as local versions of compactified Jacobians (see Beauville [4] and Piontkowski [15]), and in representation theory as homogeneous affine Springer fibers, where they were first considered by Lusztig and Smelt in [13] and then by Piontkowski [15]. The area statistic can be generalized directly. The dinv statistic and the map ζ are a bit more tricky. It is convenient to adjust the rank function on the boxes in the following way: The steps of the boundary path of a diagram D ∈ Y N,M are ranked as before. However, for d > 1 some distinct steps might have the same rank, therefore rearranging the steps of the path according to their rank is problematic. The following idea for overcoming this difficulty was suggested by François Bergeron. It also can be found in [3]. Example 2.2 Consider the diagram D ∈ Y 9,6 with the boundary path vvvvvhvhhhvvhvh (see Figure 4). Given an (+N, +M )-invariant subset ∆ one can extract d many (+n, +m)-invariant subsets from it by the following procedure: for each r ∈ {0, 1 . . . , d − 1} consider the subset in ∆ consisting of all integers congruent to r modulo d, subtract r from all these elements and then divide by d. In other words one has Note that the subsets ∆ r for r > 0 might be not 0-normalized. Note also that ∆ can be uniquely reconstructed from ∆ 0 , . . . , ∆ d−1 , so we have a bijection between 0-normalized (+N, +M )-invariant subsets and (ordered or Z/dZ-colored) collections of d many (+n, +m)-invariant subsets, such that ∆ 0 is zero normalized and ∆ i ⊂ Z ≥0 for all i.
Let us first describe this in an example. The idea of the equivalence relation is that one should fix the collection ∆ 0 , . . . , ∆ d−1 up to shifts, but allow them to shift with respect to each other if it does not change the total relative order of N -generators and M -cogenerators of ∆. By a shift of ∆, we mean replacing ∆ with ∆ + k for some k ∈ Z. It is more illustrative to split ∆ 1 into its even and odd parts: It is more compact to then stack them as r = 0 −4 −2 0 2 4 6 8 10 12 14 16 18 . . . 1 1 3 5 7 9 11 13 15 17 Then we just record ∆ 1 0 and ∆ 1 1 : −2 −1 0 1 2 3 4 5 6 7 8 9 10 . . .
Note that the sequences of N -generators and M -cogenerators are the same for ∆ 1 and ∆ 2 , even if we take into account the remainder modulo 2. In both cases one gets where red is for even generators and cogenerators (r = 0), and blue is for odd (r = 1). The reason we set ∆ 1 ∼ ∆ 2 is that in this example the odd part can be shifted by 1 with respect to the even part without changing the sequence or parity of generators and cogenerators. Note that one cannot shift further: in ∆ 1 one cannot shift the odd part to the left, and in ∆ 2 one cannot shift the odd part to the right. Also note that while The formal definition of the equivalence classes is as follows: m be the corresponding collections of (+n, +m)-invariant subsets as in (1). We say that ∆ 1 and ∆ 2 are equivalent or ∆ 1 ∼ ∆ 2 if there exist a permutation σ ∈ S d such that for each i there exists an integer k i such that ∆ 1 i = k i + ∆ 2 σ(i) and the following condition is satisfied. The sequence of Z/dZ-colored N -generators and M -cogenerators of ∆ 1 agrees with the sequence of σ-twisted colored N -generators and M -cogenerators of ∆ 2 .
Obviously, the order of N -generators and M -cogenerators is preserved within an equivalence class. The bijection to Y M,N is constructed in two steps. First, we choose the "minimal" representative in each class, and second, we construct a Young diagram for each minimal representative. Bijectivity is evident from the construction.
Definition 2.7 The skeleton of an (+N, +M )-invariant subset ∆ is the set consisting of its N -generators and M -cogenerators.
Note that one can uniquely reconstruct the invariant subset ∆ from its skeleton. Indeed, the skeleton contains all the N -generators of ∆, and to distinguish the N -generators from the M -cogenerators one should simply choose the biggest elements in each congruence class mod N. Definition 2.9 Let ∆ be a 0-normalized (+N, +M )-invariant subset and let ∆ 0 , . . . , ∆ d−1 be the corresponding collection of (+n, +m)-invariant subsets as in (1). We say that ∆ is minimal if the following conditions are satisfied: 1. For every 0 < i < d there exists j < i such that the intersection of the skeletons of ∆ i and ∆ j is nonempty; 2. If the skeleton of ∆ i has empty intersection with the skeleton of ∆ i+1 then min(∆ i ) < min(∆ i+1 ).
Theorem 2.10 Every equivalence class of (+N, +M )-invariant subsets contains a unique minimal element.
Sketch of the proof. Let ∆ ∈ M N,M . Consider its skeleton colored with respect to the remainders modulo d. Let us allow the parts of the skeleton of different colors to shift with respect to each other while preserving the distances between elements of the skeleton of the same color and not changing the total relative order of generators and co-generators. Let us fix the part of the skeleton with remainder 0 and shift the other colors to the left as much as possible.
Note that as parts of the skeleton of ∆ shift, the correspondence between colors and remainders change. Moreover, it might happen that we will get two different colors in the same remainder. Let S 0 , . . . , S d−1 ⊂ Z be the parts of the shifted skeleton of different colors. Let r i be the remainder of S i modulo d. We can require the indices i and the remainders r i to satisfy the following conditions.
1. If r i < r j , then i < j; 2. If r i = r j and min(S i ) < min(S j ), then i < j.
Note that these conditions determine the order on colors uniquely. Now, let Then the subset ∆ ∈ M N,M is the unique invariant subset with skeleton S . In other words, we reassign remainders to colors according to the refined order, while preserving the sets Si d for each color. By construction, we have ∆ ∼ ∆ . It is also not hard to see that ∆ is minimal. One can also see that if ∆ is minimal, then ∆ = ∆.
The last step in proving the main result is to construct the Young diagram D(∆) ∈ Y N,M from a minimal (+N, +M )-invariant subset ∆. The construction goes as follows. Consider the (+n, +m)invariant subsets ∆ 0 , . . . , ∆ d−1 . For each (+n, +m)-invariant subset there is a unique way to organize its skeleton into a cycle, so that on each edge as we go along the cycle we either add m or subtract n (basically, one should simply follow the boundary of the corresponding Young diagram). Consider the cycle corresponding to ∆ 0 . We know that 0 = min(∆ 0 ) is an n-generator and −m is an m-cogenerator. Let us remove the (+m)-edge −m → 0 from the cycle, so that we get a simple path which starts from 0 and ends at −m. We will start from this path and then insert the cycles corresponding to ∆ 1 , . . . , ∆ d−1 into it, so that in the end we get a path from 0 to −m which traverses M (−n)-edges and N − 1 (+m)edges. The rule is very simple: on kth move we insert the cycle corresponding to ∆ k in the last possible position, i.e. we go along the path constructed so far till its last intersection with the cycle corresponding to ∆ k , then go around the cycle, and then finish the rest of the path.
The resulting path encodes the boundary of the diagram D(∆) : the (+m)-edges correspond to vertical steps, while (−n)-edges correspond to horizontal steps, plus there is an extra vertical step in the very beginning. In fact, as we see in the example below, this path gives us the corresponding ranks of the steps in the boundary path. The cycle corresponding to ∆ 1 is 0 → 2 → −1 → 1 → −2 → 0.
The last intersection of the path with this cycle is at −2, so after merging one gets the path The cycle corresponding to ∆ 2 is 6 → 8 → 5 → 7 → 4 → 6.
The last intersection of the new path with this cycle is at 4, so the resulting path is Finally, the boundary path of D(∆) is vvvvvhvhhhvvhvh (see Figure 5). Note that in this example one could first merge ∆ 2 and then ∆ 1 with the same result. This is because their skeletons do not intersect.
Similar to the d = 1 case, the steps of the boundary path correspond to the N -generators and Mcogenerators of ∆. However, the correspondence is a bit trickier. The label in the box above a horizontal step (correspondingly, to the left of a vertical step) is equal to x d , where x is the M -cogenerator (Ngenerator). Sketch of the proof. Given a diagram D ∈ Y N,M , all we need to do to recover the corresponding minimal (+N, +M )-invariant subset is to assign the remainders modulo d to the steps of the boundary path of D. This is done by reverting the gluing procedure as follows. Find the lowest m + n consecutive steps in the boundary path of D, whose endpoints differ by − −−−− → (m, −n). Color these steps d−1. Then remove this part of the path, and shift the upper part of the remaining steps by − −−−− → (m, −n) to make the remaining path connected. Repeat this procedure, using the color d − k at the k-th iteration (see Figure 5 for an example).