1 Introduction

Rational parking functions and Catalan numbers

Abstract.

The “classical” parking functions, counted by the Cayley number , carry a natural permutation representation of the symmetric group in which the number of orbits is the Catalan number . In this paper, we will generalize this setup to “rational” parking functions indexed by a pair of coprime positive integers. We show that these parking functions, which are counted by , carry a permutation representation of in which the number of orbits is the “rational” Catalan number . We compute the Frobenius characteristic of the -module of -parking functions. Next we study -analogues of the rational Catalan numbers, proposing a combinatorial formula for and relating this formula to a new combinatorial model for -binomial coefficients. Finally, we discuss -analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.

This work was partially supported by a grant from the Simons Foundation (#244398 to Nicholas Loehr).
Third author supported in part by National Science Foundation grant DMS-1201312.

1. Introduction

1.1. Overview of Parking Functions

The goal of this paper is to generalize the theory of “classical” parking functions — counted by  — to the theory of “rational” parking functions — counted by for coprime positive integers . The combinatorial foundation for this theory is given by rational Dyck paths, which are lattice paths staying weakly above a line of slope . The classical case corresponds to , which for most purposes is equivalent to considering a line of slope .

One major algebraic motivation for studying parking functions comes from representation theory. It was conjectured by Haiman [21, Conjecture 2.1.1] that the -module of “diagonal coinvariants” has dimension and that the naturally bi-graded character of this module might be encoded by parking functions [21, Conjecture 2.6.3]. The Shuffle Conjecture [19, Conjecture 3.1.2] (which is still open) gives such a precise description. We suggest in Definition 24 a way to generalize the combinatorial form of the Shuffle Conjecture to rational parking functions (see also Conjecture 27). However, we do not know what algebraic or geometric objects might underlie the generalized combinatorics. The fact that the combinatorics works out so nicely suggests that there must be an underlying reason. See [14, 15, 23] for related work that arises from a more geometric viewpoint.

The Shuffle Conjecture and related formulas for -Catalan numbers involve certain statistics on Dyck paths, namely paired with either or . The statistic is natural and the and statistics are strange. However, there is a bijection on Dyck paths that sends to and to . Thus one could say that there is really just one natural statistic and one strange map . In this paper we will define an analogous map that we use to define -analogues of rational parking functions. Experimentally, this map has beautiful combinatorial properties. Conjecture 27 contains several conjectures pertaining to the symmetry and specializations of these rational -parking functions. In a forthcoming paper [4] we will show that our map belongs to a whole family of sweep maps that contain a wealth of combinatorial information.

We will also study -analogues of rational Catalan numbers that are closely related to the specialization mentioned above. These rational -Catalan numbers are given algebraically by where (see §1.5 for the definition of this notation). Although these expressions have long been known to be polynomials in with nonnegative integer coefficients, it is an open problem to find a combinatorial interpretation for these polynomials. We propose such a combinatorial model in §4 along with a related non-standard combinatorial interpretation for general -binomial coefficients.

1.2. Outline of the Paper

The structure of the paper is as follows. We conclude the introduction by reviewing standard notation concerning partitions, symmetric functions, and -binomial coefficients that will be used throughout the paper; for more details, consult [29, 32, 39]. §2 gives background pertaining to diagonal coinvariants and classical parking functions, deriving several formulas for the ungraded Frobenius character of these modules. §3 generalizes this discussion to the case of rational parking functions. The key to deriving the Frobenius characters of these new modules is Proposition 2, which enumerates rational parking functions with a specified vertical run structure. §4 studies the rational -Catalan numbers, proposing new combinatorial formulas for these polynomials and for general -binomial coefficients based on certain partition statistics. We prove that Conjecture 8, which proposes a novel combinatorial interpretation for -binomial coefficients, is equivalent to Conjecture 6, which proposes a combinatorial interpretation for rational -Catalan numbers. §5 reviews the classical theory of -parking functions and -Catalan numbers, culminating in the Shuffle Conjecture for the Frobenius series of the doubly-graded module of diagonal coinvariants. §6 generalizes this theory to the case of rational -Catalan numbers and rational -parking functions. Finally, §7 includes explicit computations of various polynomials considered in this paper.

1.3. Notation for Partitions

A partition is a weakly decreasing sequence of positive integers. The -th part of is . The area of is . We say that is a partition of (denoted by ) when . The length of is , the number of parts of . We consider the empty sequence to be the unique partition of ; this partition has length zero. It is sometimes convenient to add one or more zero parts to the end of a partition; this does not change the length of the partition. Let be the set of all partitions. For all , is the number of parts of equal to . The integer is defined by

For example, is a partition with , , , , and .

1.4. Notation for Symmetric Functions

Let denote the ring of symmetric functions, which we view as a subring of as in [32, Ch. I]. We now recall some bases of used later in the paper. First, the complete homogeneous symmetric functions are defined by the generating function

(1)

For any partition of length , let . Then is the complete homogeneous basis of . Second, the power sum symmetric functions are defined for all by setting . For any partition of length , let . Then is the power sum basis of . Third, the monomial symmetric function is the sum of all distinct monomials with nonzero exponents (in some order). The monomial basis of is . Finally, the Schur basis of (defined in [32]) is denoted .

1.5. Notation for -Binomial Coefficients

Let be a formal variable. For all integers , define the -integer , the -factorial , and the -binomial coefficient

(2)

We usually use the multinomial coefficient notation when discussing -binomial coefficients.

2. Classical Parking Functions

The permutations of the set are counted by the factorial . Two other important objects in combinatorics, trees and parking functions, are counted by the Cayley number . One can algebraically motivate the progression from to as follows.

2.1. Diagonal Coinvariants

The symmetric group acts on the polynomial ring by permuting variables. Isaac Newton knew that the subring of “symmetric polynomials” is generated by the power sum polynomials

Claude Chevalley [8] knew that the quotient ring of coinvariants

is isomorphic to a graded version of the regular representation of , and so has dimension . Moreover, the Hilbert series of is the -factorial:

Armand Borel [6] knew that is also the cohomology ring of the complete flag variety. It turns out that the structure of is closely related to the combinatorial structure of permutations.

More generally, acts diagonally on the polynomial ring by simultaneously permuting the -variables and the -variables. Hermann Weyl [43] knew that the subring of -invariant polynomials is generated by the polarized power sums:

Mark Haiman [21, Conjecture 2.1.1] conjectured that the quotient ring of diagonal coinvariants

has dimension as a vector space over . Furthermore, this vector space is bi-graded by -degree and -degree, and he saw hints that the bi-graded Hilbert series

is closely related to well-known structures in combinatorics. Haiman, along with Adriano Garsia, made several conjectures in this direction [11]. However, because the polarized power sums are not algebraically independent, it turned out to be quite difficult to prove these conjectures. Some have now been proved, some are still open, and in general the subject remains very active.

2.2. Parking Functions

Arthur Cayley [7] showed that there are trees with labeled vertices. However, for the purposes of studying diagonal coinvariants, we prefer to discuss parking functions [25, 37], which can be defined as follows. There are cars that want to park in spaces along a one-way street. Each car has a preferred spot and the cars park in order: first, second, etc. When car arrives it will try to park in spot . If spot is already taken it will park in the first available spot after . If no such spot exists, the parking process fails. We say that the -tuple is a classical parking function if it allows all of the cars to park.

For example, is a parking function. The order of the parked cars is shown here:

One may check that is a parking function if and only if its increasing rearrangement satisfies for all . This leads to a few observations.

First, the set of parking functions is closed under permuting subscripts. Let denote both the set of parking functions and the corresponding permutation representation of  [38, Def. 1.3.2]. Haiman conjectured that, after a sign twist, the ungraded -module of diagonal coinvariants is isomorphic to . We describe the graded version of this conjecture in §2.6 below.

Second, the increasing parking functions are in bijection with the set of Dyck paths. We define a classical Dyck path of order as a sequence in with the property that every initial subsequence has at least as many ’s as ’s. If we read from left to right, interpreting as “go north” and as “go east”, we can think of this as a lattice path in from to staying weakly above the diagonal line . Now, given an increasing parking function , let be the number of times that occurs in . Then we associate with the Dyck path

For example, the increasing parking function corresponds to the Dyck path :

The number of Dyck paths of order is the Catalan number

Third, generalizing the previous observation, we associate (possibly non-increasing) parking functions with labeled Dyck paths. Given a parking function we first draw the Dyck path corresponding to the increasing rearrangement. The -th vertical run of the path has length (possibly zero), which corresponds to the number of occurrences of in . If we have with then we will label the -th vertical run by the set of indices . We do this by filling the boxes to the right of the -th vertical run with the labels , increasing vertically. For example, the parking function corresponds to the following labeled Dyck path:

This is the standard way to encode parking functions in the literature on diagonal coinvariants. We will use this encoding to compute the structure of the -module .

2.3. Frobenius Characteristic of

We have observed that acts on parking functions by permuting subscripts. Translating this to an action on labeled Dyck paths, acts on a labeled path by permuting labels and then reordering labels in each column so that labels still increase reading up each column. From this description, we see that the orbits of this action are in bijection with Dyck paths, hence are counted by the Catalan number . Suppose and a given Dyck path has vertical runs of length (for ). Then the orbit corresponding to this Dyck path has stabilizer isomorphic to the Young subgroup , where

(3)

The number of Dyck paths with this vertical run structure is known to equal

(4)

where we define so that ; more specifically, . (Equation 4 follows, for instance, from a more general result proved in Proposition 2 below.)

Next, recall that the Frobenius characteristic map is an isomorphism from the -algebra of symmetric group representations to the -algebra . This map sends a class function to the symmetric function , and the map sends the irreducible character to the Schur function  [38, §4.7]. Let denote the image of the character of under the Frobenius characteristic map.

Each orbit of with stabilizer isomorphic to contributes a term to  [32, p. 113–114]. The number of such orbits is given by (4), and hence

(5)
(6)

Using the multinomial theorem, the generating function (1), and the fact that , we obtain the identity

(7)

2.4. Character of .

To extract information from (7), we recall the Cauchy product

(8)

The following well-known lemma shows how the Cauchy product can be used to detect dual bases of .

Lemma 1.

[32, I.4.6],[29, Thm. 10.131] Define the Hall inner product on by requiring that and be dual bases. For any two bases and of such that all and are homogeneous of degree , we have if and only if these two bases are dual with respect to the Hall inner product. In particular,

(9)

It follows from (1) and (8) that by setting of the variables equal to and the rest equal to zero, specializes to . Furthermore, one sees directly that specializes to . These facts, combined with (7) and (9), yield

Hence

Applying the inverse of the Frobenius map, this formula tells us the character of the -module . On one hand, for , is the number of parking functions in fixed by the action of . On the other hand, if has cycle type (so that is a product of disjoint cycles), the preceding formula shows that . In particular, taking to be the identity permutation in , which has cycle type and fixes all parking functions in , we see that . (There are easier ways to obtain this result, but we wanted to illustrate the power of the generating function method.)

2.5. Schur Expansion of .

The derivation of the Frobenius series of can be redone using the Schur symmetric functions instead of the power sum basis. As above, we specialize to consist of zeros along with copies of . Since is homogeneous of degree , this specialization changes into , where indicates that variables have been set to and the rest to zero. By (9),

This gives the decomposition of the parking function module into irreducible constituents. In the special case of hook shapes we can compute the coefficients explicitly (see  [39, p. 364]):

These integers are called the Schröder numbers, and they also describe the -vector of the associahedron (see Pak-Postnikov [36, formula (1–6)]). Setting , we verify that the multiplicity of the trivial representation in is the classical Catalan number:

These computations were originally done by Haiman [21], Pak-Postnikov [36], and Stanley [40].

2.6. The Graded Version of .

As mentioned earlier, it is known that the parking function module and the diagonal coinvariant ring are isomorphic as (ungraded) -modules. However, also comes with a symmetric bi-grading by -degree and -degree, and it is natural to ask whether we can explain this bi-grading in terms of . This is the content of the famous Shuffle Conjecture, described in §5 below. This section studies the simpler case where we grade by the -degree only.

There is a straightforward combinatorial construction that turns the module into a singly-graded vector space. The grading is defined on basis vectors by the statistic , which is the number of boxes fully contained between the labeled Dyck path and the diagonal . For example, the following figure shows that :

Since permuting the labels of a labeled Dyck path leaves the area unchanged, we have given the structure of a graded -module. Haiman conjectured and eventually proved [11, 21, 22] that this module is isomorphic to graded by -degree only, where denotes the sign character.

3. Rational Parking Functions

In this section we will generalize the -module of classical parking functions labeled by a single positive integer to an -module of rational parking functions, denoted , labeled by two coprime positive integers and . The classical parking functions will correspond to the case . To define we must first discuss rational Dyck paths.

3.1. Rational Dyck Paths

Given positive integers , let denote the subset of consisting of words containing copies of and copies of . We can think of such a word as a lattice path in from to by reading from left to right, interpreting as “go north” and as “go east.” An element of is called an -Dyck path if it stays weakly above the diagonal line . Let denote the set of -Dyck paths. An example of a -Dyck path is shown here:

Recall that the -Dyck paths are called “classical” and they are counted by the Catalan number . Every -Dyck path must end with an east step, and by removing this east step we obtain a canonical bijection

In the case that are coprime, the enumerative theory of Dyck paths is quite nice. In particular, it was known as early as 1954 (see Bizley [5]) that when are coprime we have

We call this number the rational Catalan number, denoted . When are coprime, we call the set of rational Dyck paths in . Observe that .

The main result of this section is a formula enumerating rational Dyck paths with a specified vertical run structure. In this setting, a “vertical run of length ” is a subword such that this subword is either preceded by or is the beginning of the entire word. The classical (i.e., non-rational) version of this formula goes back at least to Kreweras’ work in 1972 [26, Theorem 4] in the context of noncrossing partitions.

Proposition 2.

Let and be coprime positive integers. Let be nonnegative integers such that and . Then the number of -Dyck paths in with vertical runs of length is given by

Proof.

Step 1. Let be the set of paths such that has vertical runs of length for all , and ends in an east step. We show

Let denote the set of words containing copies of for each . Define by replacing each letter in a word by . Since and , we see that maps into and that ends in an east step for all . It is now clear that is a bijection of onto . Since , Step 1 is complete.

Step 2. Let . We associate a level to the -th lattice point on a path as follows. Set . For each , set

Note that the level of a lattice point is . We show that the levels for a given are all distinct. For suppose for some ; we prove that must be and must be . Let and be the lattice points reached by the -th and -th steps of . We know and ; also, and . Since , . But and are coprime, so their least common multiple is . This forces to divide and to divide . So we must have , , and , giving and as claimed.

Step 3. Define an equivalence relation on by letting iff there exist such that , , and is a cyclic shift of . We show that every equivalence class has size and contains exactly one -Dyck path. Fix and write for some . By cyclically shifting by , we obtain a list of paths (not yet known to be distinct), which are the paths in equivalent to . Suppose the list of steps in has east steps at positions . Define . Cyclically shifting by steps, where , has the effect of cyclically shifting by steps. This cyclic shift will replace the sequence of levels for by the new sequence

(10)

If is the minimum level in , it follows that is the minimum level in . Since are distinct by Step 2, it follows that the paths all have distinct minimum levels, hence these paths must be pairwise distinct. Furthermore, the minimum level in must occur at the end of an east step, so for some . For every , the minimum level in (10) is , which is nonnegative iff . But since is the minimum level in and all levels are distinct, the minimum level in is nonnegative iff . This means that exactly one of the paths in the equivalence class of is an -Dyck path.

Step 4. Step 3 shows that decomposes into a disjoint union of -element subsets, each of which meets in exactly one point. So the cardinality can be found by dividing the multinomial coefficient in Step 1 by . ∎

3.2. Rational Parking Functions

An -parking function is an -Dyck path together with a labeling of the north steps by the set such that labels increase in each column going north. For example, here is a -parking function.

The symmetric group acts on -parking functions by permuting labels and then reordering the labels within columns if necessary. Let denote the set, and also the -module, of -parking functions. Consistent with our terminology for -Dyck paths, we call the set of rational parking functions in the case where and are coprime. We make this assumption of coprimality for the rest of this section. In this case, we can compute the Frobenius characteristic of by the same method used in §2.3 to compute the Frobenius characteristic of (which is essentially ).

Theorem 3.

The Frobenius characteristic of the -module of parking functions has the following expansions in terms of complete homogeneous, power sum, and Schur symmetric functions, respectively:

Proof.

The -expansion follows from Proposition 2 in the same way that (5) followed from (4). (Note that the hypothesis that and are coprime is necessary here.) By the multinomial theorem, is the coefficient of in the generating function . The power sum and Schur expansions then follow from (8) and (9), by the same arguments used in §2.4 and §2.5. ∎

Recall that the -expansion tells us the character of the permutation action of on . We can express this as follows.

Corollary 4.

Let be a permutation with cycles. Then the number of elements of fixed by equals . In particular, taking to be the identity permutation in (which has cycles), .

Using [39, page 364], we obtain the following formula for the coefficient of when is a hook shape.

Corollary 5.

For , the multiplicity of the hook Schur function in is

We call these integers the rational Schröder numbers, denoted . We note that these numbers also occur as the -vector of the recently discovered rational associahedron [3].

We already knew a special case of this result. Namely, when , we find that the multiplicity of the trivial character in is the rational Catalan number

But since is a permutation module, the multiplicity of the trivial character is the number of orbits. Since the orbits are represented by -Dyck paths, we recover Bizley’s result regarding the number of -Dyck paths [5]. This result can also be proved by an argument similar to the one in Proposition 2; see [29, §12.1].

Observe from Corollary 5 that the sign character of occurs in if and only if . More precisely, we see that the smallest value of for which occurs in is .

3.3. Graded Version of .

It is straightforward to generalize the statistic on classical Dyck paths to rational Dyck paths. For coprime and , is the number of boxes fully contained between and the diagonal line . For example, the rational Dyck path displayed at the beginning of §3.2 has area 5. The action of on preserves the statistic, so this statistic turns into a singly-graded -module, as in the classical case.

When and are coprime, the maximum value of over all -Dyck paths is . Indeed, the diagonal of the rectangle intersects a ribbon of boxes, as shown here:

Note that this ribbon divides the rest of the rectangle into two equal pieces of size

4. Rational -Catalan Numbers

This section studies a -analogue of the rational Catalan number obtained by replacing by . We conjecture combinatorial interpretations for these polynomials, as well as a new combinatorial formula for all -binomial coefficients, based on certain partition statistics. The main theorem of this section shows that the conjecture for -binomial coefficients implies the conjecture for rational -Catalan numbers.

4.1. Background on -Binomial Coefficients.

The -binomial coefficients defined by (2) are in fact polynomials in . One can prove this algebraically by checking that the -binomial coefficients satisfy the recursions

(11)

and initial conditions .

Recall the following well-known combinatorial interpretation of -binomial coefficients. Let denote the set of partitions whose diagrams fit in the box with corners , , , and . More precisely, iff where the parts are integers satisfying . Note that is the number of boxes in the diagram of . One may verify that

(12)

by showing that the right side of (12) satisfies the recursions in (11) (see, e.g., [29, §6.7]). We refer to (12) as the standard combinatorial interpretation for the -binomial coefficients, to distinguish it from the non-standard interpretation presented below.

4.2. Rational -Catalan Numbers

Suppose and are positive integers with . The rational -Catalan number for the slope is

(13)

It is known that is a polynomial with coefficients in . Haiman gives an algebraic proof of this fact in [21, Prop. 2.5.2] and also gives an algebraic interpretation for these polynomials as the Hilbert series of a suitable quotient ring of a polynomial ring  [21, Prop. 2.5.3 and 2.5.4]. This polynomial also appears to be connected to certain modules arising in the theory of rational Cherednik algebras. Our purpose here is to propose a combinatorial interpretation for the rational -Catalan numbers. More precisely, given with , our problem is to find a set of combinatorial objects and a statistic