Cyclic sieving and rational Catalan theory

Let $a<b$ be coprime positive integers. Armstrong, Rhoades, and Williams defined a set $\mathsf{NC}(a,b)$ of `rational noncrossing partitions', which form a subset of the ordinary noncrossing partitions of $\{1, 2, \dots, b-1\}$. Confirming a conjecture of Armstrong et. al., we prove that $\mathsf{NC}(a,b)$ is closed under rotation and prove an instance of the cyclic sieving phenomenon for this rotational action. We also define a rational generalization of the $\mathfrak{S}_a$-noncrossing parking functions of Armstrong, Reiner, and Rhoades.


Introduction
This paper is about generalized noncrossing partitions arising in rational Catalan combinatorics. A set partition of [n] := {1, 2, . . . , n} is noncrossing if its blocks do not cross when drawn on a disk whose boundary is labeled clockwise with 1, 2, . . . , n. Noncrossing partitions play a key role in algebraic and geometric combinatorics. Along with an ever-expanding family of other combinatorial objects, the noncrossing partitions of For coprime positive integers a and b, the rational Catalan number is Cat(a, b) = 1 a+b a+b a,b . Observe that Cat(n, n + 1) = Cat(n) and Cat(n, mn + 1) = Cat (m) (n), so that rational Catalan numbers are a further generalization of Fuss-Catalan numbers. Inspired by favorable representation theoretic properties of the rational Cherednik algebra attached to the symmetric group S a at parameter b a , the research program of rational Catalan combinatorics seeks to further generalize Catalan combinatorics to the rational setting.
Some rational generalizations of Catalan objects have been around for decades -the rational analog of a Dyck path dates back at least to the probability literature of the 1940s. Armstrong, Rhoades, and Williams used rational Dyck paths to define rational analogs of polygon triangulations, noncrossing perfect matchings, and noncrossing partitions [2]. This paper goes deeper into the study of rational noncrossing partitions.
For coprime parameters a < b, Armstrong et. al. defined the a, b-noncrossing partitions to be a subset NC(a, b) of the collection of noncrossing partitions of [b − 1] arising from a laser construction involving rational Dyck paths (see Section 2 for details). It was shown that NC(a, b) is counted by Cat(a, b), as it should be, and that NC(n, mn + 1) is the set of m-divisible noncrossing partitions of [mn], as it should be.
However, the construction of NC(a, b) in [2] was indirect and involved the intermediate object of rational Dyck paths. This left open the question of whether many of the fundamental properties of classical noncrossing partitions generalize to the rational case. For example, it was unknown whether the set NC(a, b) is closed under the dihedral group of symmetries of the disk with b − 1 labeled boundary points. Consequently, the rich theory of counting noncrossing set partitions fixed by a dihedral symmetry (see [12]) lacked a rational extension. Moreover, the unknown status of rotational closure made it difficult to generalize the noncrossing parking functions of Armstrong, Reiner, and Rhoades [1] (or the 2-noncrossing partitions of Edelman [6]) to the rational setting. The core problem was that the natural dihedral symmetries of noncrossing partitions are harder to visualize on the level of Dyck paths, even in the classical case.
The purpose of this paper is to resolve the issues in the last paragraph to support NC(a, b) as the 'correct' definition of the rational noncrossing partitions. We will prove the following.
• The action of rotation on NC(a, b) exhibits a cyclic sieving phenomenon generalizing that for the action of rotation on classical noncrossing partitions (Theorem 5.3). • The numerology of partitions in NC(a, b) with a nontrivial rotational symmetry generalizes that of classical noncrossing partitions with a nontrivial rotational symmetry (Corollaries 4. 10, 4.11, 4.12). • Partitions in NC(a, b) can be decorated to obtain a S a × Z b−1 -set of rational noncrossing parking functions Park N C (a, b). The formula for the permutation character of this set generalizes the corresponding formula for the classical case (Theorem 6.3).
The key to obtaining the rational extensions of classical results presented above will be to develop a better understanding of the set NC(a, b). We will give two new characterizations of this set (Propositions 3.3 and 3.15). The more important of these will involve an idea genuinely new to rational Catalan combinatorics: a new measure of size for blocks of set partitions in NC(a, b) called rank.
In the Fuss-Catalan case (a, b) = (n, mn + 1), the rank of a block is determined by its cardinality (Proposition 3.6). However, rank and cardinality diverge at the rational level of generality. The main heuristic of this paper is that: The rank of a block of a rational noncrossing partition is a better measure of its size than its cardinality.
In Section 3, we will prove that rank shares additivity and dihedral invariance properties with size. We will prove that ranks (unlike cardinalities) characterize which noncrossing set partitions of [b − 1] lie in NC(a, b). In Section 6, ranks (not cardinalities) will be used to define and study rational noncrossing parking functions. In Sections 4 and 5, ranks (not cardinalities) will be used to give rational analogs of enumerative results for noncrossing partitions with rotational symmetry.
2. Background 2.1. Rational Dyck paths. The prototypical object in rational Catalan combinatorics is the rational Dyck path. Let (a, b) be coprime positive integers. An a, b-Dyck path (or just a Dyck path when a and b are clear from context) is a lattice path in Z 2 consisting of north and east steps which starts at (0, 0) ends at (b, a), and stays above the line y = a b x. The 5, 8-Dyck path N EN EEN N EN EEE is shown in Figure 1. When (a, b) = (n, n + 1), rational Dyck paths are equivalent to classical Dyck paths -lattice paths from (0, 0) to (n, n) which stay weakly above y = x.
If D is an a, b-Dyck path, a valley of D is a lattice point P on D such that P is immediately preceded by an east step and succeeded by a north step. A vertical run of D is a maximal contiguous sequence of north steps; the number of vertical runs equal the number of valleys. The 5, 8-Dyck path shown in Figure 1 has 4 valleys. The vertical runs of this path have sizes 1, 1, 2, and 1.
The numerology associated to rational Dyck paths generalizes that of classical Dyck paths. The number of a, b-Dyck paths is the rational Catalan number Cat(a, b) = 1 a+b a+b a,b . The set of a, b-Dyck paths D with k vertical runs is counted by the rational Narayana number Given a length a vector of nonnegative integers r = (r 1 , r 2 , . . . , r a ) satisfying ir i = a, the number of a, b-Dyck paths with r i vertical runs of size i for 1 ≤ i ≤ a is given by the rational Kreweras number where k = r i is the total number of vertical runs. For example, the 5, 7-Dyck path shown in Figure 2 contributes to Krew(5, 7; r), where r = (1, 2, 0, 0, 0).

Noncrossing partitions.
A set partition π of [n] is called noncrossing if, for all indices 1 ≤ i < j < k < ≤ n, we have that i ∼ k in π and j ∼ ∈ π together imply that i ∼ j ∼ k ∼ in π. Equivalently, the set partition π is noncrossing if and only if the convex hulls of the blocks of π do not intersect when drawn on the disk with boundary points labeled clockwise with 1, 2, . . . , n.
We let NC(n) denote the collection of noncrossing partitions of [n]. The rotation operator rot acts on the index set [n], the power set 2 [n] , and the collection NC(n) by the permutation 1 2 . . . n − 1 n 2 3 . . . n 1 . These three sets also carry an action of the reflection operator rfn by the permutation 1 2 . . . n − 1 n n n − 1 . . . 2 1 . Together, rot and rfn generate a dihedral action on these sets.

2.3.
Rational noncrossing partitions. In [2], rational Dyck paths were used to construct a rational generalization of the noncrossing partitions. Let D be an a, b-Dyck path and let P = (0, 0) be a lattice point which is at the bottom of a north step of D. The laser (P ) is the line segment of slope a b which is 'fired' northeast from P and continues until it intersects the Dyck path D. By coprimality, the east endpoint of (P ) is necessarily on the interior of an east step of D.
Let D be an a, b-Dyck path. We define a set partition π(D) of [b − 1] as follows. Label the east ends of the non-terminal east steps of D from left to right with 1, 2, . . . , b − 1 and fire lasers from all of the valleys of D. The set partition π(D) is defined by the 'visibility' relation i ∼ j if and only if the labels i and j are not separated by laser fire. (Here we consider labels to lie slightly below their lattice points.) By construction, the set partition π(D) is noncrossing.
An example of this construction when (a, b) = (5, 8) is shown in Figure 1. If D is an a, b-Dyck path, we have a natural 'visibility' bijection from the set of vertical runs of D to the set of blocks of π(D) which associates a vertical run at y = i to the block of π(D) whose minimum element is i + 1. The visibility bijection is shown with colors in Figure 1. It will be convenient to think of the lasers fired from the valleys of a Dyck path D in terms of their endpoints. We let the laser set (D) of the a, b-Dyck path D be the set of pairs (i, j) such that D contains a laser starting from a valley with x-coordinate i and ending in the interior of an east step with west x-coordinate j. For the 5, 8-Dyck path D shown in Figure 1, we have that (3,7), (4, 5)}.
For a < b coprime, we define the set of admissible lasers By considering π(D) for all possible a, b-Dyck paths D, we get the set of a, b-noncrossing partitions NC(a, b) := {π(D) : D an a, b-Dyck path} (This is called the set of inhomogeneous a, b-noncrossing partitions in [2].) It is clear from construction that NC(a, b) ⊆ NC(b − 1). Some basic facts about a, b-noncrossing partitions are as follows.
Proposition 2.1. Let a < b be coprime positive integers.
(1) The map π : {a, b-Dyck paths} → NC(a, b) is bijective, so that |NC(a, b)| = Cat(a, b) and the number of a, b-noncrossing partitions with k blocks is the rational Narayana number Nar(a, b; k).
When (a, b) = (n, n + 1), the set NC(n, n + 1) of rational noncrossing partitions is just the set NC(n) of all noncrossing partitions of [n]. When (a, b) = (n, mn + 1) we have that NC(n, mn + 1) of rational noncrossing partitions is the set of all noncrossing partitions of [mn] which are m-divisible in the sense that every block size is divisible by m. Armstrong, Rhoades, and Williams posed the problem of finding an analogous 'intrinsic' characterization of NC(a, b) for arbitrary a < b coprime. We give two such characterizations in Section 3.

2.4.
Homogeneous rational noncrossing partitions. Rational Dyck paths are used in [2] to construct a generalization of noncrossing perfect matchings on the set [2n]. The construction is similar to that of the rational noncrossing partitions.
Let (a, b) be coprime and let D be an a, b-Dyck path. We construct a set partition π(D) of the set [a + b − 1] as follows. Label the interior lattice points of D from southwest to northeast with 1, 2, . . . , a + b − 1. Fire lasers of slope a b from every lattice point of D which is at the south end of a north step (not just those lattice points which are valleys of D). The set partition π(D) is given by declaring i ∼ j if and only if the labels i and j are not separated by laser fire. As before, we consider labels to be slightly below their lattice points. Topological considerations make it clear that π(D) is a noncrossing partition of [a + b − 1]. An example of this construction is shown in Figure 2 for (a, b) = (5,8).
Considering π(D) for all possible a, b-Dyck paths D gives rise to the set of a, b-homogeneous rational noncrossing partitions The adjective 'homogeneous' refers to the fact that every set partition in HNC(a, b) has a blocks. By construction, we have that HNC(a, b) ⊆ NC(a + b − 1).
When (a, b) = (n, n + 1), the set HNC(n, n + 1) is the set of noncrossing perfect matchings on [2n]. When (a, b) = (n, mn + 1), the set HNC(n, mn + 1) is the set of noncrossing set partitions on [mn] in which every block has size m (these are also called m-equal noncrossing partitions). Some basic facts about HNC(a, b) for general a < b are as follows.
Proposition 2.2. Let a < b be coprime positive integers.
(1) The map The set HNC(a, b) is closed under the rotation operator rot.
In [2] a promotion operator on a, b-Dyck paths is shown to intertwine the action of rot with the map π from paths to homogeneous noncrossing partitions. We will give an inhomogeneous analog of this promotion operator in the next section.

Characterizations of Rational Noncrossing Partitions
3.1. The action of rotation. Let a < b be coprime. In this subsection we will prove that NC(a, b) is closed under the rotation operator rot by describing rotation (or, rather, its inverse) as an operator on a, b-Dyck paths. We will define an operator The definition of rot is probably best understood visually. Figure 3 shows three 11, 13-Dyck paths. The middle path is the image of the left path under rot and the right path is the image of the middle path under rot . The valley lasers are fired on each of these Dyck paths, and the corresponding partitions in NC (11,13) are shown below, but the vertex labels on the Dyck paths are omitted for legibility.
Let D 1 be the 11, 13-Dyck path on the left of Figure 3. The westernmost horizontal run of D 1 has size > 1. Because of this, we define D 2 := rot (D 1 ) by removing one step from this horizontal run and adding it to the easternmost horizontal run of D 1 . The lattice path D 2 is displayed in the middle of Figure 3. Informally, the path D 2 is obtained from D 1 by translating one unit west.
Let D 2 be the 11, 13-Dyck path in the middle of Figure 3. The westernmost horizontal run of D 2 has size 1. The definition of D 3 := rot (D 2 ) is more complicated in this case. We break the lattice path D 2 up into four subpaths. The first subpath, shown in black, is the initial northern run of D 2 . The second subpath, shown in red, is the single east step which occurs after this run. The third subpath, shown in green, extends from the westernmost valley of D 2 to the (necessarily east) step just before the laser fired from this valley hits D 2 . The fourth subpath, shown in blue, extends from the end of the third subpath to the terminal point (b, a) of D 2 . The path D 3 is obtained by  concatenating the third, first, fourth, and second subpaths, in that order. The concatenation of green, then black, then blue, then red is shown on the right of Figure 3. More formally, given an a, b-Dyck path D, the definition of rot (D) breaks up into three cases. Let D = N i 1 E j 1 · · · N im E jm be the decomposition of D into nonempty vertical and horizontal runs.
(2) If m, j 1 > 1, we set (3) If m > 1 and j 1 = 1, let P = (1, i 1 ) be the westernmost valley of D. The laser (P ) fired from P hits D on a horizontal run E i k for some 2 ≤ k ≤ m. Suppose that (P ) hits the horizontal run E i k on step r, where 2 ≤ r ≤ i k . We set Proposition 3.1. The definition of rot given above gives a well defined operator on the set of a, b-Dyck paths. As operators {a, b-Dyck paths } −→ NC(a, b), we have that rot −1 • π = π • rot . In particular, the set NC(a, b) is closed under rotation.
Proof. Let D be an a, b-Dyck path. If the westernmost horizontal run of D has size > 1, it is clear that rot (D) is also an a, b-Dyck path and that the corresponding set partitions are related by rot −1 (π(D)) = π(rot (D)). We therefore assume that the westernmost horizontal run of D consists of a single step. We claim that the lattice path rot (D) stays above the diagonal y = a b x. Indeed, consider the decomposition of rot (D) as in Figure 3. The first (green) subpath of rot (D) stays above y = a b x because the laser fired from the westernmost valley of D has slope a b . The second (black) subpath of rot (D) is a vertical run, so the concatenation of the first and second subpaths of rot (D) stay above y = a b x. The third (blue) subpath of rot (D) is just the corresponding subpath of D translated one unit west, and certainly stays above the line y = a b x. Since the fourth (red) subpath of rot (D) is a single east step, we conclude that the entire path rot (D) stays above y = a b x, and so is an a, b-Dyck path. By the last paragraph, the set partition π(rot (D)) ∈ NC(a, b) is well defined. We argue that rot −1 (π(D)) = π(rot (D)). To do this, we consider how the valley lasers of rot (D) relate to the corresponding valley lasers of D. of rot (D) between its first (green) and second (black) subpaths, which necessarily hits rot (D) on its terminal east step. From this description of the valley lasers of rot (D), one checks that rot −1 (π(D)) = π(rot (D)), as desired.

3.2.
Characterization from Kreweras complement. Our first characterization of rational noncrossing partitions gives a description of their Kreweras complements. Let π be a noncrossing partition of [n]. The Kreweras complement krew(π) is the noncrossing partition obtained by drawing the 2n vertices 1, 1 , 2, 2 , . . . , n, n clockwise on the boundary of a disk in that order, drawing the blocks of π on the unprimed vertices, and letting krew(π) be the unique coarsest partition of the primed vertices which introduces no crossings. The map krew : NC(n) → NC(n) satisfies krew 2 = rot, and so is a bijection.
An example of Kreweras complementation for n = 7 is shown in Figure 4. We have that ) but its Kreweras complement (the all singletons set partition) is not. On the other hand, the set NC(a, b) is in bijective correspondence with its Kreweras image krew(NC(a, b)). The problem of characterizing NC(a, b) is therefore equivalent to the problem of characterizing its Kreweras image.
Given π ∈ NC(a, b), there is a simple relationship between the blocks of krew(π) ∈ NC(b − 1) and the laser set (D) of the a, b-Dyck path D corresponding to π. Lemma 3.2. Let a < b be coprime, let π ∈ NC(a, b) have corresponding a, b-Dyck path D, and let krew(π) be the Kreweras complement of π. The laser set (D) is B is a block of krew(π) and i is a nonmaximal element of B}.
Proof. This is clear from the topological definition of π(D) and Kreweras complement.
Let π ∈ NC(b − 1) be a noncrossing partition and suppose we would like to determine whether π is in NC(a, b). In light of Lemma 3.2, it is enough to know whether there exists an a, b-Dyck path D whose laser set consists of pairs (i, max(B)), where i ∈ B runs over all nonmaximal elements of all blocks B of krew(π). A characterization of the laser sets of a, b-Dyck paths can be read off from the results of [13]. In [13] this characterization was used to prove that the rational analog of the associahedron is a flag simplicial complex. Proposition 3.3. Let a < b be coprime and let π be a noncrossing partition of [b − 1]. We have that π is an a, b-noncrossing partition if and only if for every block B of the Kreweras complement krew(π) we have that • (i, max(B)) ∈ A(a, b) for every nonmaximal element i ∈ B, and • for any two nonmaximal elements i < j in B, Proof. As mentioned above, we know that π ∈ NC(a, b) if and only if there exists an a, b-Dyck path D whose laser set consists of all possible pairs (i, max(B)), where B ranges over all blocks of krew(π) and i ranges over all nonmaximal elements of B. For such an a, b-Dyck path D to exist, it is certainly necessary that (i, max(B)) ∈ A(a, b) always, so we assume this condition holds.  To determine which of π 1 , π 2 , π 3 belong to NC(5, 8), we start by computing their Kreweras images: Since both of the bullets in Proposition 3.3 hold for π 3 , we conclude that π 3 ∈ NC(5, 8).
3.3. a, b-ranks. In order to state our second characterization of a, b-noncrossing partitions, we introduce a new way to measure the 'size' of a block B of a noncrossing partition π other than its cardinality |B|. Our rational analog of block size is as follows.
Definition 3.5. Let a < b be coprime positive integers and let π ∈ NC(b − 1) be a noncrossing partition of [b − 1]. We assign an integer rank π a,b (B) ∈ Z to every block B of π (or just rank(B) when a, b, and π are clear from context) by the following recursive procedure. Let be the partial order on the blocks of π defined by The integers rank π a,b (B) are implicitly determined by the formula for all blocks B ∈ π.
While we define rank π a,b (·) as a function on the blocks of an arbitrary noncrossing partition of [b − 1], this notion of rank will be more useful for a, b-noncrossing partitions. We will see that rank is more useful than cardinality as a block size measure in rational Catalan theory. This subsection proves basic properties of the function rank π a,b (·) on the blocks of a, b-noncrossing partitions. In the Fuss-Catalan case b ≡ 1 (mod a), rank and cardinality are equivalent.
Definition 3.7. If π ∈ NC(a, b) is an a, b-noncrossing partition, the rank sequence R(π) = (r 1 , . . . , r b−1 ) of π is the sequence of nonnegative integers given by For example, the rank sequence of the 5, 8-noncrossing partition shown in Figure 1 has rank sequence (1, 1, 0, 2, 1, 0, 0). Observe that this is also the sequence of vertical runs of the corresponding 5, 8-Dyck path, so that the rank sequence is equivalent to the Dyck path. This is a general phenomenon.
Proposition 3.8. Let a < b be coprime, let π be an a, b-noncrossing partition with rank sequence R(π) = (r 1 , . . . , r b−1 ), and let D be the a, b-Dyck path associated to π.
(1) For any block B of π, the vertical run of D visible from π has size rank(B).
(2) The Dyck path D is given by Proof. For any block B of π, we have that min(B) labels the lattice point just to the right of the vertical run labeled by B. Therefore, (2) follows from (1).
To prove (1), recall the partial order on the blocks of π. If B is minimal with respect to , then B = [min(B), max(B)] is an interval. The labels of B must appear on a single horizontal run of D, say on the line y = c. Let i B be the length of the vertical run visible from B, let P be the valley at the bottom of this vertical run, and let (P ) be the laser fired from P .
The laser (P ) must intersect the line y = c. We claim that it does so in the open x-interval (max(B), max(B) + 1). Indeed, by coprimality there exists a unique lattice point Q on D which is northwest of the laser (P ) and has minimum horizontal distance to the laser (P ). Let m be the x-coordinate of Q. We claim that m ∈ B. Indeed, if m / ∈ B, there must be a laser (Q ) fired from a valley Q which separates m from B. But then Q would be closer than Q to (P ), so we conclude that m ∈ B. Since B = [min(B), max(B)] is an interval, we have that (P ) intersects y = c in the open x-interval (max(B), max(B) + 1). This implies that rank(B) = i B , as desired. Now suppose that B is not -minimal among the blocks of π. Then the interval [min(B), max(B)] is a union of at least two blocks of π. For any block B contained in this interval, let i B denote the size of the vertical run visible from B . We may inductively assume that, for all B = B, we have that rank π a,b (B ) = i B . Let π 0 be the set partition obtained from π by merging the blocks contained in [min(B), max(B)]. Then π 0 is noncrossing, and hence a, b-noncrossing by Proposition 2.1 (2). The recursion for rank says that where the second equality used the inductive hypothesis. Moreover, the Dyck path D 0 corresponding to π 0 is obtained from D by replacing the portion between the x-coordinates min(B) and max(B) with a single vertical run, followed by a single horizontal run. In particular, the size of the vertical The cardinality function | · | on blocks of noncrossing partitions satisfies the following two properties.
Recall that rot acts by adding 1 to every index, modulo b − 1. Rotation invariance is therefore must each be unions of blocks of π. Let B 1 , . . . , B r be a complete list of these remaining blocks. We certainly have rank π a,b (B i ) = rank On the other hand, Proposition 3.1 and the assumption that π ∈ NC(a, b) guarantee that rot(π) ∈ NC(a, b). Proposition 3.8 shows that (rot(B)).
By Proposition 3.8, every a, b-noncrossing partition has a valid a, b-ranking. It would be nice if the converse held, but it does not; if (a, b) = (2, 5), the partition π = {{1, 2, 4}, {3}} has a valid 2, 5-ranking but π / ∈ NC(2, 5). Our second characterization of a, b-noncrossing partitions states that π ∈ NC(a, b) if and only if every element in the orbit of π under rotation has a valid a, b-ranking.
To prove this statement, we start by examining how validity behaves under coarsening.
In general, having a valid a, b-ranking is not closed under coarsening of noncrossing partitions. For example, let (a, b) = (5, 11) and consider the two partitions π, π ∈ NC(10) given by Then π has a valid 5, 11-ranking but π does not. On the other hand, certain types of coarsening do preserve validity.
Lemma 3.14. Let a < b be coprime and let π ∈ NC(b − 1) be a noncrossing partition such that π has a valid a, b-ranking. Suppose that π ∈ NC(b − 1) is another noncrossing partition obtained from π by one of the following two operations: (1) merging two blocks B 0 , B 1 of π with min(B 1 ) = max(B 0 ) + 1, or (2) merging two blocks B 0 , B 1 of π with B 0 ≺ B 1 . The noncrossing partition π has a valid a, b-ranking.
Observe that if π, π ∈ NC(b − 1) are such that π refines π and π is a union of intervals, then π can be obtained from π by a sequence of coarsenings as described in Lemma 3.14.
Proof. First assume that π is obtained from π by replacing B 0 and B 1 with B 0 ∪B 1 as in Condition 1. For i = 0, 1, let r i denote the sum of the ranks of the blocks B of π satisfying B ≺ B i . By the definition of rank, we have a b for i = 0, 1. Adding these two equations together and recalling that x + y −1 ≤ x+y ≤ x + y for any x, y ∈ R, we get Since π has a valid a, b-ranking, we have rank π a,b (B i ) > 0 for i = 0, 1, so that rank π a,b (B 0 ∪ B 1 ) > 0. Our analysis breaks up into two cases depending on whether B 0 ∪ B 1 is a -maximal block of π .
If B 0 ∪ B 1 is a -maximal block of π , for every block B of π with B = B 0 , B 1 we have rank π a,b (B) = rank π a,b (B), so that the chain of inequalities in (9) implies B∈π rank π a,b (B) ≤ a, so B∈π rank π a,b (B) = a by Lemma 3.13. Since every block of π has a positive rank, we conclude that π has a valid a, b-ranking.
If B 0 ∪ B 1 is not a -maximal block of π , there exists a unique block B 2 ∈ π such that B 2 covers B 0 ∪ B 1 in . The recursion for rank π a,b (·) gives where r 2 is the sum of the ranks of all of the blocks B of π satisfying B ≺ B 2 but B B 0 , B 1 . Combining (10) with the inequalities in (9), we see that rank π a,b (B 2 ) ≥ rank π a,b (B 2 ) > 0. Since we have rank π a,b (B) = rank π a,b (B) for all blocks B = B 0 , B 1 , B 2 . In particular, the rank of every block of π is positive and B∈π rank π a,b (B) = a. We conclude that π has a valid a, b-ranking. Now assume that π is obtained from π by replacing B 0 and B 1 with B 0 ∪ B 1 as in Condition 2. It follows that B 1 covers B 0 in the -order. By the recursion for rank, (12) rank π a,b (B 0 ∪ B 1 ) = rank π a,b (B 0 ) + rank π a,b (B 1 ) and the ranks of all other blocks of π equal the corresponding ranks in π , so that π has a valid a, b-ranking.
We are ready to prove our a, b-rank characterization of a, b-noncrossing partitions. Proof. Suppose π ∈ NC(a, b). By Proposition 3.1, we know that the rotation orbit of π is contained in NC(a, b), so that every partition in this orbit has a valid a, b-ranking.
For the converse, suppose that π ∈ NC(b − 1) − NC(a, b). We argue that some partition in the rotation orbit of π does not have a valid (a, b)-ranking.
Consider the Kreweras complement krew(π). If there is a block B ∈ krew(π) and an index i ∈ B − {max(B)} such that (i, max(B)) / ∈ A(a, b), then π refines the two-block set partition π : consists of two intervals, we have that rot −i (π ) can be obtained from rot −i (π) by a sequence of coarsenings as in Lemma 3.14. The condition (i, max(B)) / ∈ A(a, b) means that so that rot −i (π ) does not have a valid a, b-ranking. By Lemma 3.14, we conclude that rot −i (π) does not have a valid a, b-ranking. By the last paragraph, we may assume that for every block B ∈ krew(π) and every index i ∈ B −{max(B)}, we have (i, max(B)) ∈ A(a, b). Since π ∈ NC(b−1)−NC(a, b), by Proposition 3.3 there exists a block B ∈ krew(π) and indices i, j ∈ B − {max(B)} with i < j such that where max(B) = k. The set partition π refines the three-block noncrossing partition π : consists of three intervals, we have that rot −i (π ) can be obtained from rot −i (π) by a sequence of coarsenings as in Lemma 3.14. We show that rot −i (π ) does not have a valid a, b-ranking; by Lemma 3.14, this implies that rot −i (π) does not have a valid a, b-ranking and completes the proof. Working towards a contradiction, suppose π := rot −i (π ) has a valid a, b-ranking. Denote the blocks of π by On the other hand, by Lemma 3.14, we have that π := {B 1 ∪ B 2 , B 3 } also has a valid a, b-ranking. Moreover, we have that rank π a,b (B 3 ) = rank π a,b (B 3 ). This implies that Putting these facts together gives a = rank π a,b (B 1 ) + rank π a,b (B 2 ) + rank π a,b (B 3 ) which is a contradiction. We conclude that π does not have a valid a, b-ranking.
As an application of Proposition 3.15, we get that the set NC(a, b) carries an action of not just the rotation operator rot, but the full dihedral group rot, rfn . This gives another application of a, b-ranks. Proof. It is enough to show that NC(a, b) is closed under rfn. For any noncrossing partition π ∈ NC(b − 1) and any block B of π, we have that rank π a,b (B) = rank rfn(π) a,b (rfn(B)).
It follows that every element in the rotation orbit of π has a valid a, b-ranking if and only if every element in the rotation orbit of rfn(π) has a valid a, b-ranking.

Modified rank sequences
In this section we will study rational noncrossing partitions which have nontrivial rotational symmetry. Our key tool will the the theory of a, b-rank developed in Section 3. We fix the following Notation. For the remainder of this section, let a < b be coprime positive integers. Let d|(b − 1) be a divisor with 1 ≤ d < b − 1. The numerology of NC d (a, b) will turn out to be somewhat simpler than that of NC(a, b) itself. We begin by defining a modified version of the rank sequence which is well suited to studying partitions with rotational symmetry.
Let π ∈ NC d (a, b). The fact that π is noncrossing implies that π has at most one block B 0 which satisfies rot d (B 0 ) = B 0 . If such a block B 0 exists, it is called central. Moreover, the cyclic group Z q = rot d acts freely on the non-central blocks of π. A non-central block B of π is called wrapping if the interval [min(B), max(B)] contains every block in the rot d -orbit of B. Any rot d -orbit of blocks has at most one wrapping block.  Let π ∈ NC d (a, b). The d-modified rank sequence of π is the length d sequence S d (π) = (s 1 , . . . , s d ) of nonnegative integers defined by (13) s i := rank π a,b (B) if i = min(B) for some non-central, non-wrapping block B ∈ π, 0 otherwise.
The d-modified rank sequence S d (π) is like the ordinary rank sequence R(π), but we only consider the indices in [d] rather than in [b−1] and we only keep track of the ranks of blocks which are neither central nor wrapping. It is true, but not obvious at this point, that a set partition π ∈ NC d (a, b) is determined by S d (π).
Our first lemma states that the assignment π → S d (π) commutes with the action of rotation.
Proof. Let S d (rot(π)) = (s 1 , s 2 , . . . , s d ) be the d-modified rank sequence of rot(π) and let 1 ≤ i ≤ q. We show that s i = s i−1 , where subscripts are interpreted modulo d. We will make free use of Proposition 3.10, which implies that rank π a,b (B) = rank rot(π) a,b (rot(B)) for any block B ∈ π. Case 1: 2 ≤ i ≤ d. Suppose s i−1 > 0. Then i − 1 = min(B) for some non-central, non-wrapping block B ∈ π. Since B is non-central and non-wrapping and 1 ≤ min(B) ≤ d − 1, we know that rot(B) is also non-central and non-wrapping with min(rot(B)) = i. We conclude that s i = s i−1 .
Suppose s i−1 = 0. If i − 1 is not the minimum element of a block of π, then i is not the minimum element of a block of rot(π), so that s i = 0. If i − 1 = min(B 0 ) for a central block B 0 ∈ π, then rot(B 0 ) is a central block in rot(π) with i ∈ rot(B 0 ), so that s i = 0. If i − 1 = min(B) for a wrapping block B ∈ π, then the fact that 1 ≤ min(B) ≤ d − 1 implies that either i = min(rot(B)) or (rot(B) is wrapping with i ∈ rot(B)). In either situation, we get that s i = 0.
Case 2: i = 1. Suppose s d > 0. Then d = min(B) for some non-central, non-wrapping block B ∈ π. Recalling that rot d (π) = π, it follows that rot d(q−1)+1 (B) is a non-central, non-wrapping block of rot(π) containing 1. Thus, we get s 1 = rank rot(π) a,b (rot d(q−1)+1 (B)) = rank π a,b (B) = s q . Suppose s d = 0. If d is contained in a central block of π, then 1 is contained in a central block of rot(π) and s 1 = 0. Since π is noncrossing and rot d (π) = π, the index q cannot be contained in a wrapping block of π. If d ∈ B for some block B ∈ π which is non-central and non-wrapping, we must have that d = min(B). Since π is noncrossing with rot d (π) = π, it follows that rot(B) is wrapping and 1 ∈ rot(B), so that s 1 = 0.
What sequences (s 1 , . . . , s d ) of nonnegative integers arise as d-modified rank sequences of partitions in NC d (a, b)? If π ∈ NC d (a, b) and S d (π) = (s 1 , . . . , s d ) is the d-modified rank sequence of π, we claim that where the sum is over all non-central blocks B ∈ π. Indeed, each q-element orbit of non-central blocks contributes the rank of one of its constituents precisely once to the nonzero terms in S d (π). By Proposition 3.15, with equality if and only if π does not have a central block. (Unless q|a, the partition π necessarily has a central block.) We call a length d sequence (s 1 , . . . , s d ) of nonnegative integers good if we have the inequality s 1 + · · · + s d ≤ a q . The goal for the remainder of this section is to show that the map is a bijection. Since good sequences are easily enumerated, this will give us information about NC d (a, b). The strategy is to isolate nice subsets of NC d (a, b) and the set of good sequences which contain at least one representative from every rotation orbit, show that these subsets are in bijection under S d , and apply Lemma 4.2.  . . , s q ) is a very good sequence, let L(s) be the lattice path obtained by taking the q-fold concatenation of the path N s 1 E . . . N s d E, adding a terminal east step, and if s 1 + · · · + s d < a q adding an initial vertical run of size c := a − q(s 1 + · · · + s d ). In symbols, q . Since s is assumed to be very good, we get that L(s) ends at (b, a) so that the map L is well defined. We will refer to the subpaths L 1 , . . . , L q defined by the above factorization of L(s) as the segments of L(s), so that L(s) = L 1 · · · L q E.
Lemma 4.6. Every good sequence is rot -conjugate to at least one noble sequence.
Case 1: c = a. In this case (s 1 , . . . , s d ) is the zero sequence (0, 0, . . . , 0) and is trivially noble. Case 2: 0 < c < a. The argument we present here is a modification of the argument used to prove the Cycle Lemma.
Consider the lattice path L which starts at the origin and ends at (2d, 2(s 1 + · · · + s d )) given by As is common in rational Catalan theory, we label the lattice points P on L with integers w(P ) as follows. The origin is labeled 0. Reading L from left to right, if P and P are consecutive lattice points, we set w(P ) = w(P ) − a if P is connected to P with an E-step and w(P ) = w(P ) + b if P is connected to P with an N -step. For example, suppose that (a, b) = (11,13), d = 4, and (s 1 , s 2 , s 3 , s 4 ) = (1, 0, 2, 0). We have that q = 13 4 = 3 and c = 11 − 3(1 + 0 + 2 + 0) = 2. The lattice path L, together with the labels of its lattice points, is as follows. By coprimality, there exists a unique lattice point on L of minimal weight. Let P 0 be this lattice point; in the above example, we have w(P 0 ) = −14. We claim that P 0 occurs after a pair of consecutive east steps EE. Indeed, since 0 < c < a, we know that the weight of the terminal lattice point of L is negative (in the above example, −10), so that P 0 is not the origin. If P 0 did not occur after a pair of consecutive east steps, by the minimality of w(P 0 ), we have that P 0 occurs after a pair N E. But since a < b the lattice point P 0 occurring at the beginning of this N E-sequence satisfies w(P 0 ) = w(P 0 ) + a − b < w(P 0 ), a contradiction. Therefore, the lattice point P 0 does indeed occur after a pair of consecutive east steps EE.
The minimality of w(P 0 ) and the fact that (s 1 , . . . , s d ) = (0, . . . , 0) mean that P 0 is immediately followed by a nonempty vertical run N s i for some 1 ≤ i ≤ d. Since P 0 is preceded by a pair EE, we get that i ≥ 2 and s i−1 = 0. In the above example, we have that i = 3. Since s i−1 = 0, the rotated sequence (s i−1 , s i , . . . , s d , s 1 , s 2 , . . . , s i−2 ) is very good. The lattice path L(s i−1 , s i , . . . , s d , s 1 , s 2 , . . . , s i−2 ) is therefore well defined.
We claim that the lattice path L(s i−1 , s i , . . . , s d , s 1 , s 2 , . . . , s i−2 ) is always an a, b-Dyck path, so that the rotation (s i−1 , s i , . . . , s d , s 1 , s 2 , . . . , s i−2 ) is a noble sequence. Indeed, consider the segmentation L(s) = L 1 . . . L q E. The segments L 1 , . . . , L q will be progressively further east, so it is enough to show that the final segment L q stays west of the line y = x. By construction, the segment L q starts with a single east step, hits the copy of the point P 0 , then then has a nonempty vertical run, and eventually ends at the point (b − 1, a). Since (s 1 , . . . , s d ) is a good sequence, we know that L q starts at a lattice point to the west of the line y = a b (x + 1). The minimality of w(P 0 ) forces L q to remain west of the line y = a b x. Case 3: c = 0. This is a special case of the Cycle Lemma; the argument is very similar to Case 2 and is only sketched.
We again consider the lattice path L given by L = (N s 1 E . . . N s d E)(N s 1 E . . . N s d E) and assign weights w(P ) to the lattice points P on L as before. There exists a unique lattice point P 0 on L with minimal weight. The lattice point P 0 necessarily occurs before a nonempty vertical run N s i for some 1 ≤ i ≤ d. The rotation (s i , s i+1 , . . . , s d , s 1 , s 2 , . . . , s i−1 ) of s is a noble sequence.
Given any noble sequence s, we may consider the a, b-noncrossing partition π(L(s)) corresponding to the Dyck path L(s). We prove that π(L(s)) is rot d -invariant and noble.
Proof. Recall the visibility bijection between blocks of π and nonempty vertical runs in L(s). Let c = a − q(s 1 + · · · + s d ). The argument depends on whether c > 0 or c = 0.
Case 1: c > 0. We consider the segmentation of L(s) given by As an example of this case, consider (a, b) = (11,13), d = 4, s = (0, 2, 0, 1). Then s is a noble sequence. The 11, 13-Dyck path L(s) = L 1 L 2 L 3 E is shown below, where vertical hash marks separate the segments L 1 , L 2 , and L 3 . The valley lasers from L(s) are as shown. For visibility, we suppress the labels on the interior lattice points.
Since s is noble and c > 0, we have s 1 = 0. Fix any index 2 ≤ i ≤ d such that s i > 0 and another index 1 ≤ j ≤ q − 1. Both of the segments L j and L j+1 of the a, b-Dyck path L(s) contain a copy of the nonempty vertical run N s i . If P 0 and P 1 are the valleys at the bottom of these vertical runs in L j and L j+1 , respectively, we have that the lasers (P 0 ) and (P 1 ) fired from P 0 and P 1 are (rigid) translations of the same line segment. In particular, the block visible from the copy of N s i in L j+1 is the image of the block visible from the copy of N s i in L j by the operator rot d . Moreover, the fact that s 1 = 0 means that neither of these blocks contain the index 1.
In general, we conclude that the set of blocks of π not containing 1 is stable under the action of rot d , so that the block of π containing 1 must be central. We get that π ∈ NC d (a, b). Since π ∈ NC d (a, b) has a central block containing 1, the partition π contains no wrapping blocks. It follows that π in noble and S d (π) = s.
Case 2: c = 0. In this case, π does not contain a central block. We again consider the segmentation L(s) = L 1 L 2 . . . L q E as in Case 1. Here L j = N s 1 EN s 2 E . . . N s d E for 1 ≤ j ≤ q.
As an example of this case, consider (a, b) = (9, 13), d = 4, and s = (1, 2, 0, 0). We have that s is a noble sequence. The 9, 13-Dyck path L(s) is shown below, with diagonal hash marks denoting the segmentation L(s) = L 1 L 2 L 3 E. The valley lasers of L(s) are shown, and the interior lattice point labels are suppressed.
Let 1 ≤ i ≤ d be an index such that s i > 0 and consider any two consecutive segments L j and L j+1 of L(s). The lasers fired from the valleys below the copies of N s i both L j and L j+1 are either • translates of each other, or • both hit the Dyck path L(s) = L 1 . . . L q E on its terminal east step.
In the above example, we see that the lasers fired from each copy of N s 2 = N 2 are translates of one another, and the lasers fired from each copy of N s 1 = N 1 hit L(s) on its terminal east step. This implies that the blocks corresponding of these vertical runs are rot d -conjugate, so that π ∈ NC d (a, b).
Moreover, since c = 0 the laser fired from any copy of the (nonempty) vertical run N s 1 in L(s) = L 1 . . . L q E must hit L(s) on its terminal east step. This implies that π has no wrapping blocks, so that π is noble and S d (π) = s.
We show that nobility for sequences and nobility for partitions coincide. a, b). Then π is noble if and only if S d (π) is noble.
(⇒) Suppose π is noble. Since π contains no wrapping blocks, the rank sequence R(π) is given by the formula If c > 0, since 1 is contained in the central block of π we get s 1 = 0, so that S d (π) is very good. By Proposition 3.8, we get that L(s) is an a, b-Dyck path, so that S d (π) is noble.
(⇐) Suppose S d (π) is noble. We claim that π contains no wrapping blocks. Working towards a contradiction, assume that π contains at least one wrapping block and choose a wrapping block B of π such that the interval [min(B), max(B)] is maximal under containment.
If 1 ∈ B, then we would have s 1 = 0 since B is wrapping, making the first step of the path L(S d (π)) an east step. This contradicts the nobility of S d (π). We conclude that 1 / ∈ B. Since 1 / ∈ B, the interval [1, min(B) − 1] is nonempty. By our choice of B, we have that [1, min(B) − 1] is a union blocks of π, none of which are central or wrapping. This means that the first min(B) − 1 terms in the d-modified rank sequence S d (π) coincide with the first min(B) − 1 terms in the ordinary rank sequence R(π).
By Proposition 3.8, the a, b-Dyck path corresponding to π starts at the origin with the subpath N s 1 EN s 2 E . . . N s min(B)−1 E. Since [1, min(B) − 1] is a union of blocks of π, Corollary 3.9 guarantees that this subpath ends at the point (min(B) − 1, a b (min(B) − 1) ). It follows that the subpath (N s 1 EN s 2 E . . . N s min(B)−1 E)E obtained by appending a single east step crosses the diagonal y = a b x. But since B is wrapping, we have s min(B) = 0, so that this is an initial subpath of L(S d (π)), so that L(S d (π)) is not an a, b-Dyck path. This contradicts the nobility of S d (π). We conclude that π contains no wrapping blocks.
Suppose that π contains a central block B 0 . We need to prove that 1 ∈ B 0 . If 1 / ∈ B 0 , the fact that π does not contain wrapping blocks implies that [1, min(B 0 ) − 1] is a union of nonwrapping, non-central blocks of π. The first min(B 0 )−1 terms of the d-modifed rank sequence S d (π) coincide with the corresponding terms of the ordinary rank sequence R(π). The same reasoning as in the last paragraph implies that the lattice path L(S d (π)) contains the point (min(B 0 ) − 1, a b (min(B 0 ) − 1) ). However, since B 0 is central, we have that s min(B 0 ) = 0, so that the lattice path L(S d (π)) has an east step originating from this lattice point. But this means that L(S d (π)) is not an a, b-Dyck path, contradicting the nobility of S d (π). We conclude that 1 ∈ B 0 , and that π is a noble partition.
We have the lemmata we need to prove that the map S d is a bijection. In particular, partitions in NC d (a, b) are determined by their d-modified rank sequences. Proof. By Lemma 4.2, we know that S d commutes with rotation. If s is a noble sequence, Lemma 4.7 shows that S −1 d (s) is nonempty, and Lemma 4.6 shows that S d is surjective. To see that S d is injective, let s be a noble sequence and suppose π ∈ NC d (a, b) satisfies S d (π) = s = (s 1 , . . . , s d ). By Lemma 4.8, the partition π is noble. The rank sequence R(π) is therefore R(π) = (s 1 , . . . , s d , s 1 , . . . , s d , . . . , s 1 , . . . , s d ) if s 1 + · · · + s d = a q , (c, . . . , s d , s 1 , . . . , s d , . . . , s 1 , . . . , s d ) if s 1 + · · · + s d < a q , where c = a − q(s 1 + · · · + s d ). By Proposition 3.8, any a, b-noncrossing partition is determined by its rank sequence. We conclude that |S −1 d (s)| = 1. By Observation 4.4 and Lemma 4.6, together with the fact that S d commutes with rot (Lemma 4.2), we conclude that S d is a bijection.
Since every nonzero entry in a d-modified rank sequence S d (π) = (s 1 , . . . , s d ) corresponds to a rot d -orbit of non-central blocks of π of that rank, the following result follows immediately from Proposition 4.9. We can also consider counting partitions in NC d (a, b) with a fixed number of non-central block orbits (of any rank). By Proposition 4.9, this is equivalent to counting sequences (s 1 , . . . , s d ) of nonnegative integers with bounded sum and a fixed number of nonzero entries.
Proof. For the first part, we choose p entries in the sequence (s 1 , . . . , s d ) to be nonzero. Then, we assign positive values to these p entries in such a way that their sum is < ad b−1 . The second part is similar.
Finally, we can consider the problem of counting NC d (a, b) itself. By Proposition 4.9, this corresponds to counting sequences (s 1 , . . . , s d ) of nonnegative integers which satisfy s 1 +· · ·+s d ≤ ad b−1 .

Cyclic sieving
Let X be a finite set, let C = c be a finite cyclic group acting on X, let X(q) ∈ N[q] be a polynomial with nonnegative integer coefficients, and let ζ ∈ C be a root of unity with multiplicative order |C|. The triple (X, C, X(q)) exhibits the cyclic sieving phenomenon (CSP) if, for all d ≥ 0, we have X(ζ d ) = |X c d | = |{x ∈ X : c d .x = x}| (see [12]). In this section we prove cyclic sieving results for the action of rotation on NC(a, b).
Our proofs will be 'brute force' and use direct root-of-unity evaluations of q-analogs. We will make frequent use of the following fact: If x ≡ y (mod z), then From this, we get the useful fact that Theorem 5.1. Let a < b be coprime and let r = (r 1 , r 2 , . . . , r a ) be a sequence of nonnegative integers satisfying a i=1 ir i = a. Set k := a i=1 r i . Let X be the set of a, b-noncrossing partitions of [b − 1] with r 1 blocks of rank 1, r 2 blocks of rank 2, . . . , and r a blocks of rank a.
The triple (X, C, X(q)) exhibits the cyclic sieving phenomenon, where C = Z b−1 acts on X by rotation and is the q-rational Kreweras number.
Proof. Reiner and Sommers proved that the q-Kreweras number Krew q (a, b, r) is polynomial in q with nonnegative integer coefficients using algebraic techniques [11]. No combinatorial proof of the polynomiality or the positivity of Krew q (a, b, r) is known.
We have that X(ζ d ) = 0 unless at t|r i for all but at most one 1 ≤ i ≤ a, and that r i 0 ≡ 1 (mod t) if t r i 0 . If the sequence r satisfies the condition of the last sentence, define a new sequence (m 1 , . . . , m a ) by m i = r i t for 1 ≤ i ≤ a. Let m = m 1 + · · · + m a . Write r i 0 = c i 0 t + s i 0 for s i 0 ∈ {0, 1} and assume t|r i for all i = i 0 . We have By Corollary 4.10, we have X(ζ d ) = |X rot d |.
The following Narayana version of Theorem 5.1 proves a CSP involving the action of rotation on a, b-noncrossing partitions with a fixed number of blocks.
Theorem 5.2. Let a < b be coprime, let 1 ≤ k ≤ a, and let X be the set of (a, b)-noncrossing The triple (X, C, X(q)) exhibits the cyclic sieving phenomenon, where C = Z b−1 acts on X by rotation and Proof. Reiner and Sommers proved that the q-Narayana numbers Nar q (a, b, k) are polynomials in q with nonnegative integer coefficients [11]. As in the Kreweras case, no combinatorial proof of this fact is known.
Let ζ = e 2πi b−1 and let d|(b − 1) with 1 ≤ d < b − 1. Let q = b−1 d . By Corollary 4.11, it is enough to show that otherwise. The argument here is similar to that in the proof of Theorem 5.1 and is left to the reader. The next CSP was asked for in [2, Subsection 6.2]. Theorem 5.3. Let a < b be coprime and let X be the set of (a, b)-noncrossing partitions of The triple (X, C, X(q)) exhibits the cyclic sieving phenomenon, where C = Z b−1 acts on X by rotation and is the q-rational Catalan number.
Proof. Let ζ = e 2πi b−1 and let d|(b − 1) with 1 ≤ d < b − 1. By Corollary 4.12 it is enough to show that The argument here is similar to that in the proof of  6. Parking functions 6.1. Noncrossing parking functions. Let W be an irreducible real reflection group with Coxeter number h. Armstrong, Reiner, and Rhoades defined a W × Z h -set Park N C W called the set of Wnoncrossing parking functions [1]. Given a Fuss parameter m ≥ 1, a Fuss extension Park N C W (m) of Park N C W was defined in [14]. An increasingly strong trio of conjectures (Weak, Intermediate, and Strong) was formulated about these objects and it was shown that the weakest of these uniformly implies various facts from W -Catalan theory which are at present only understood in a case-by-case fashion.
In this section, we give a rational extension of the constructions in [1,14] when W = S a is the symmetric group. This gives evidence that NC(a, b) gives the 'correct' definition of rational noncrossing partitions. Extending the work of [1,14] to other reflection groups remains an open problem.
Definition 6.1. Let a < b be coprime. An a, b-noncrossing parking function is a pair (π, f ) where π ∈ NC(a, b) is an a, b-noncrossing partition and B → f (B) is a labeling of the blocks of π with subsets of [a] such that • we have [a] = B∈π f (B), and • for all blocks B ∈ π we have |f (B)| = rank π a,b (B). We denote by Park N C (a, b) the set of all a, b-noncrossing parking functions. By Proposition 3.6, when (a, b) = (n, mn + 1), the set Park N C (n, mn + 1) agrees with the construction of Park N C Sn (m) given in [14]. In the classical case (a, b) = (n, n+1), the set Park N C (n, n+ 1) appeared in the work of Edelman under the name of '2-noncrossing partitions' [6]. The set Park N C (a, b) carries an action of S a × Z b−1 .
Proposition 6.2. The set Park N C (a, b) carries an action of the product group S a × Z b−1 , where S a acts by label permutation and Z b−1 acts by rotation.
Proof. We need to know that rotation preserves a, b-ranks of blocks, which is Proposition 3.10.
In order to state a formula for the character of the action in Proposition 6.2, we will need some notation. Let V = C a / (1, . . . , 1) be the reflection representation of S a and let ζ = e 2πi b−1 . Given w ∈ S a and d ≥ 0, let mult w (ζ d ) be the multiplicity of ζ d as an eigenvalue in the action of w on V . With this notation, a formula for the character χ is given by the following formula. Theorem 6.3. Let w ∈ S a and let g be a generator of Z b−1 . We have that for all w ∈ S a and d ≥ 0.
The multiplicity mult w (ζ d ) can be read off from the cycle structure of w. Namely, for d|b − 1 we have that is a rational extension of the Weak Conjecture of [1,14] when W = S a is the symmetric group.
The argument used to prove [14,Proposition 8.6] can be combined with the enumerative results of Section 4 to prove Theorem 6.3. We quickly illustrate how this is done.
Proof. (of Theorem 6.3) Let (w, g d ) ∈ S a × Z b−1 . We want to show that χ(w, g d ) = b multw(ζ d ) . Without loss of generality, we may assume that d|b − 1. Let q := b−1 d . The argument depends on whether q = 1 or q > 1.
Case 1: q = 1. In this case, we are ignoring the action of Z b−1 and considering Park N C (a, b) as an S a -set. We construct an S a -equivariant bijection from Park N C (a, b) to another S a -set which is known to have the correct character.
Let Park a,b be the set of all sequences (p 1 , p 2 , . . . , p a ) of positive integers whose nondecreasing rearrangement (p 1 ≤ p 2 ≤ · · · ≤ p a ) satisfies p i ≤ b a (i − 1) + 1. Equivalently, the histogram with left-to-right heights (p 1 − 1, p 2 − 1, . . . , p a − 1) stays below the line y = b a x. Sequences in Park a,b are called rational slope parking functions.
The symmetric group S a acts on Park a,b by w.(p 1 , p 2 , . . . , p a ) := (p w(1) , p w(2) , . . . , p w(a) ). It is known that the character of this action is given by Equation 19 with ζ = 1.
We build an S a -equivariant bijection ϕ : Park N C (a, b) ∼ − → Park a,b as follows. Let (π, f ) be an a, b-noncrossing parking function. Define a sequence (p 1 , p 2 , . . . , p a ) by letting p i = min(B), where B is the unique block of π satisfying i ∈ f (B). Proposition 3.8 guarantees that (p 1 , p 2 , . . . , p a ) is a sequence in Park a,b , so that the assignment ϕ : (π, f ) → (p 1 , p 2 , . . . , p a ) gives a well defined function ϕ : Park N C (a, b) → Park a,b . It is clear that ϕ is S a -equivariant. Moreover, if ϕ(π, f ) = multinomial coefficient d m 1 ,m 2 ,...,ma,d−tσ . With π fixed, we consider how to build the labeling f of the blocks of π. The labeling f must pair off the rot d -orbits of non-central blocks of π of rank i with the non-singleton w-orbits of blocks of σ of size i. For every i, there are m i ! ways to do this matching. As each of these orbits has size q, we also have q ways to rotate f within each orbit after this matching is chosen. In summary, we have that the number of pairs (π, f ) ∈ Park N C (a, b) (g,w d ) satisfying τ (π, f ) = σ is q m 1 · · · q ma m 1 ! · · · m a ! d m 1 , . . . , m a , d − t σ = q m 1 · · · q ma d!
Applying Equation 22, we obtain Equation 21, completing the proof. as an affine complex space. We refer the reader to [15] for the definitions of the objects in the following result. The proof of Theorem 6.4 is almost a word-for-word recreation of [15,Sections 4,5]. One need only replace the reference to the proof of [14,Lemma 8.5] in the proof of [15,Lemma 4.6] with the corresponding argument the fifth paragraph of Case 2 in the proof of Theorem 6.3 (which ultimately relies on Corollary 4.10).

Closing Remarks
This paper has focused entirely on rational Catalan theory for the symmetric group. The more ambitious problem of extending rational Catalan combinatorics to other reflection groups W is almost entirely open. However, the results of this paper give a roadmap for defining rational noncrossing partitions for the hyperoctohedral group.
Let W (B n ) denote the hyperoctohedral group of signed permutations of [n]. In the classical and Fuss-Catalan cases, objects associated to the group W (B n ) are obtained by considering those attached to the 'doubled' symmetric group S 2n which are invariant under antipodal symmetry. When (a, d) → (2n, b−1 2 ), the formulas in Corollaries 4.10, 4.11, and 4.12 reduce to the hyperoctohedral analogs of the rational Kreweras, Narayana, and Catalan numbers (here we view 2n as the Coxeter number of W (B n ) and let the rational parameter b be coprime to 2n). Thus, restricting to objects with antipodal symmetry gives the correct numerology for type B, even in the rational setting. It would be interesting to see how far the techniques of this paper can be extended to develop on rational Catalan combinatorics outside of type A.

Acknowledgements
The authors are grateful to Drew Armstrong and Vic Reiner for helpful conversations. B. Rhoades was partially supported by NSF grants DMS -1068861 and DMS -1500838.