The representation of the symmetric group
on -Tamari intervals
An -ballot path of size is a path on the square grid consisting of north and east unit steps, starting at , ending at , and never going below the line . The set of these paths can be equipped with a lattice structure, called the -Tamari lattice and denoted by , which generalizes the usual Tamari lattice obtained when . This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of variables. The representation of the symmetric group on these spaces is conjectured to be closely related to the natural representation of on (labelled) intervals of the -Tamari lattice, which we study in this paper.
An interval of is labelled if the north steps of are labelled from 1 to in such a way the labels increase along any sequence of consecutive north steps. The symmetric group acts on labelled intervals of by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of . In particular, the dimension of the representation, that is, the number of labelled -Tamari intervals of size , is found to be
These results are new, even when .
The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of -Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables and , a derivative with respect to and iterated divided differences with respect to . The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with “catalytic” variables.
Key words and phrases:Enumeration — Representations of the symmetric group — Lattice paths — Tamari lattices — Parking functions
2000 Mathematics Subject Classification:05A15, 05E18, 20C30
1. Introduction and main result
An -ballot path of size is a path on the square grid consisting of north and east unit steps, starting at , ending at , and never going below the line . It is well-known that there are
such paths , and that they are in bijection with -ary trees with inner nodes.
François Bergeron recently defined on the set of -ballot paths of size an order relation. It is convenient to describe it via the associated covering relation, exemplified in Figure 1.
Let and be two -ballot paths of size . Then covers if there exists in an east step , followed by a north step , such that is obtained from by swapping and , where is the shortest factor of that begins with and is a (translated) -ballot path.
It was shown in  that this order endows with a lattice structure, which is called the -Tamari lattice of size . When , it coincides with the classical Tamari lattice [6, 12, 26, 27]. Figure 2 shows two of the lattices . The main result of  gives the number of intervals in as
The lattices are also known to be EL-shellable .
The interest in these lattices is motivated by their — still conjectural — connections with trivariate diagonal coinvariant spaces [5, 8]. Some of these connections are detailed at the end of this introduction. In particular, it is believed that the representation of the symmetric group on these spaces is closely related to the representation of the symmetric group on labelled -Tamari intervals. The aim of this paper is to characterize the latter representation, by describing explicitly its character.
So let us define this representation and state our main result. Let us call ascent of a path a maximal sequence of consecutive north steps. An -ballot path of size is labelled if the north steps are labelled from 1 to , in such a way the labels increase along ascents (see the upper paths in Figure 3). These paths are in bijection with -parking functions of size , in the sense of [34, 35]: the function associated with a path satisfies if the north step of labelled lies at abscissa . The symmetric group acts on labelled -ballot paths of size by permuting labels, and then reordering them in each ascent (Figure 3, top paths). The character of this representation, evaluated at a permutation of cycle type , is
This formula is easily proved using the cycle lemma . As recalled further down, this representation is closely related to the representation of on diagonal coinvariant spaces in two sets of variables.
Now an -Tamari interval is labelled if the upper path is labelled. The symmetric group acts on labelled intervals of by rearranging the labels of as described above (Figure 3). We call this representation the -Tamari representation of . Our main result is an explicit expression for its character , which was conjectured by Bergeron and the third author .
Let be a partition of and a permutation of having cycle type . Then for the -Tamari representation of ,
Since acts by permuting labelled intervals, this is also the number of labelled -Tamari intervals left unchanged under the action of . The value of the character only depends on the cycle type , and will sometimes be denoted .
In particular, the dimension of the representation, that is, the number of labelled -Tamari intervals of size , is
We were unable to find a bijective proof of these amazingly simple formulas. Instead, our proof uses generating functions and a recursive construction of intervals. Our main generating function records the numbers , that is, the number of pairs where is a labelled interval fixed by the permutation . This generating function involves variables (keeping track of the cycle type of ) and (for the size of ). The recursive construction of intervals that we use is borrowed from . In order to translate it into an equation defining our generating function, we need to keep track of one more parameter defined on , using an additional variable (Proposition 5, Section 3). The resulting equation involves discrete derivatives (a.k.a. divided differences) with respect to , of unbounded order. The solution of equations with discrete derivatives of bounded order is now well-understood  (such equations are for instance involved in the enumeration (1) of unlabelled -Tamari intervals). But this is the first time we meet an equation of unbounded order, and its solution is the most difficult and original part of the paper. Our approach requires to introduce one more variable , and a derivative with respect to it. Its principles are explained in Section 4, and exemplified with the case . The general case is solved in Section 5. This approach was already used in a preprint by the same authors , where the special case (3) was proved. Since going from (3) to (2) implies a further complexification, this preprint may serve as an introduction to our techniques. The present paper is however self-contained. Section 6 gathers a few final comments. In particular, we reprove the main result of  giving the number of unlabelled intervals of .
In the remainder of this section, we recall some of the conjectured connections between Tamari intervals and trivariate diagonal coinvariant spaces. They seem to parallel the (now largely proved) connections between ballot paths and bivariate diagonal coinvariant spaces, which have attracted considerable attention in the past 20 years [14, 17, 18, 21, 24, 23, 29] and are still a very active area of research today [1, 2, 13, 16, 22, 19, 25, 28].
Let be a matrix of variables. The diagonal coinvariant space is defined as the quotient of the ring of polynomials in the coefficients of by the ideal generated by constant-term free polynomials that are invariant under permuting the columns of . For example, when , denoting and , the ideal is generated by constant-term free polynomials such that for all ,
An m-extension of the spaces is of great importance here [15, p. 230]. Let be the ideal of generated by alternants under the diagonal action described above; that is, by polynomials such that . There is a natural action of on the quotient space . Let us twist this action by the power of the sign representation : this gives rise to spaces
where is the -parking representation of , that is, the representation on -ballot paths of size defined above.
In the case of three sets of variables, Bergeron and Préville-Ratelle  conjecture that, as representations of ,
where is the -Tamari representation of . The fact that the dimension of this space seems to be given by (3) is an earlier conjecture due to F. Bergeron. This was also observed earlier for small values of by Haiman  in the case .
2. The refined Frobenius series
2.1. Definitions and notation
Let be a commutative ring and an indeterminate. We denote by (resp. ) the ring of polynomials (resp. formal power series) in with coefficients in . If is a field, then denotes the field of rational functions in . This notation is generalized to polynomials, fractions and series in several indeterminates. We denote by bars the reciprocals of variables: for instance, , so that is the ring of Laurent polynomials in with coefficients in . The coefficient of in a Laurent polynomial is denoted by .
We use classical notation relative to integer partitions, which we recall briefly. A partition of is a non-increasing sequence of integers summing to . We write to mean that is a partition of . Each component is called a part. The number of parts or length of the partition is denoted by . The cycle type of a permutation is the partition of whose parts are the lengths of the cycles of . This partition is denoted by . The number of permutations having cycle type equals where , where is the number of parts equal to in .
We let be an infinite list of independent variables, and for a partition of , we let . The reader may view the ’s as power sums in some ground set of variables (see e.g. ). This point of view is not really needed in this paper, but it explains why we call our main generating function a refined Frobenius series. Throughout the paper, we denote by the field of rational fractions in the ’s with rational coefficients.
Given a Laurent polynomial in a variable , we denote by the non-negative part of in , defined by
The definition is then extended by linearity to power series whose coefficients are Laurent polynomials in . We define similarly the positive part of , denoted by .
We now introduce several series and polynomials which play an important role in this paper. They depend on two independent variables and . First, we let be the following Laurent polynomial in :
We now consider the following series:
It is is a formal power series in whose coefficients are Laurent polynomials in over the field . Finally we define the two following formal power series in :
As shown with these series, we often do not denote the dependence of our series in certain variables (like and above). This is indicated by the symbol .
2.2. A refined theorem
As stated in Theorem 2, the value of the character is the number of labelled intervals fixed under the action of , and one may see (2) as an enumerative result. Our main result is a refinement of (2) where we take into account two more parameters, which we now define. The first parameter is the number of contacts of the interval: A contact of a ballot path is a vertex of lying on the line , and a contact of a Tamari interval is a contact of the lower path . We denote by the number of contacts of .
By definition of the action of on -Tamari intervals, a labelled interval is fixed by a permutation if and only if stabilizes the set of labels of each ascent of . Equivalently, each cycle of is contained in the label set of an ascent of . If this holds, we let be the number of cycles of occurring in the first ascent of : this is our second parameter.
The main object we handle in this paper is a generating function for pairs , where is a permutation and is a labelled -Tamari interval fixed by . In this series , pairs are counted by the size of (variable ), the number of contacts (variable ), the parameter (variable ), and the cycle type of (one variable for each cycle of size in ). Moreover, is an exponential series in . That is,
where the first and second sums are taken respectively over all labelled -Tamari intervals , and over all permutations fixing .
Note that when , we have:
since the number of intervals fixed by a permutation depends only on its cycle type, and since is the number of permutations of cycle type . Hence, in representation theoretic terms, is the Frobenius characteristic of the -Tamari representation of , also equal to
where is the Schur function of shape and is the multiplicity of the irreducible representation associated with in the -Tamari representation [32, Chap. 4]. For this reason, we call a refined Frobenius series.
The most general result of this paper is a (complicated) parametric expression of , which takes the following simpler form when .
Let be the refined Frobenius series of the -Tamari representation, defined by (7). Let and be two indeterminates, and write
with . In particular, in the limit , we obtain
The form of this theorem is reminiscent of the enumeration of unlabelled -Tamari intervals [8, Thm. 10], for which finding the “right” parametrization of the variables and was an important step in the solution. This will also be the case here.
Theorem 2 will follow from Theorem 3 by extracting the coefficient of in (via Lagrange’s inversion). Our expression of is given in Theorem 21. When , it takes a reasonably simple form, which we now present. The case is also detailed at the end of Section 5 (Corollary 22).
1. It is easily seen that the case of (11) reduces to the case of (9) (the proof relies on the fact that and are respectively the constant term and the positive part of in , and that is symmetric in and ).
2. When and for , the only permutation that contributes in (7) is the identity. We are thus simply counting labelled -Tamari intervals, by their size (variable ), the number of contacts (variable ) and the size of the first ascent (variable ). Still taking , we have , and the extraction of the positive part in in (11) can be performed explicitly:
When , that is, , the double sums in this expression reduce to simple sums, and the generating function of labelled Tamari intervals, counted by the size and the height of the first ascent, is expressed in terms of Bessel functions:
3. A functional equation
The aim of this section is to establish a functional equation satisfied by the series .
For , let be the refined Frobenius series of the -Tamari representation, defined by (7). Then
is the following divided difference operator
and the powers and mean that the operators are applied respectively times and times.
The above equations rely on a recursive construction of labelled -Tamari intervals. Our description of the construction is self-contained, but we refer to  for several proofs and details.
3.1. Recursive construction of Tamari intervals
We start by modifying the appearance of 1-ballot paths. We apply a 45 degree rotation to transform them into Dyck paths. A Dyck path of size consists of steps (up steps) and steps (down steps), starts at , ends at and never goes below the -axis. We say that an up step has rank if it is the up step of the path. We often represent Dyck paths by words on the alphabet . An ascent is thus now a maximal sequence of steps.
Consider an -ballot path of size , and replace each north step by a sequence of north steps. This gives a 1-ballot path of size , and thus, after a rotation, a Dyck path. In this path, for each , the up steps of ranks are consecutive. We call the Dyck paths satisfying this property -Dyck paths, and say that the up steps of ranks form a block. Clearly, -Dyck paths of size (i.e., having blocks) are in one-to-one correspondence with -ballot paths of size .
We often denote by , rather than , the usual Tamari lattice of size . Similarly, the intervals of this lattice are called Tamari intervals rather than 1-Tamari intervals. As proved in , the transformation of -ballot paths into -Dyck paths maps on a sublattice of .
Proposition 6 ([8, Prop. 4]).
The set of -Dyck paths with blocks is the sublattice of consisting of the paths that are larger than or equal to . It is order isomorphic to .
We now describe a recursive construction of (unlabelled) Tamari intervals, again borrowed from . Thanks to the embedding of into , it will also enable us to describe recursively -Tamari intervals, for any value of , in the next subsection.
A Tamari interval is pointed if its lower path has a distinguished contact. Such a contact splits into two Dyck paths and , respectively located to the left and to the right of the contact. The pointed interval is proper if is not empty, i.e., if the distinguished contact is not . We often use the notation to denote a pointed Tamari interval. The contact is called the initial contact.
Let be a pointed Tamari interval, and let be a Tamari interval. Construct the Dyck paths
as shown in Figure 4. Then is a Tamari interval. Moreover, the mapping is a bijection between pairs formed of a pointed Tamari interval and a Tamari interval, and Tamari intervals of positive size. Note that is proper if and only if the first ascent of has height larger than .
1. To recover , , , and from and , one proceeds as follows: is the part of that follows the first return of to the -axis; this also defines unambiguously. The path is the suffix of having the same size as . This also defines unambiguously. Finally, is the part of that follows the first return of to the -axis, and this also defines unambiguously.
2. This proposition is obtained by combining Proposition 5 in  and the case of Lemma 9 in . With the notation and used therein, the above paths and are respectively the parts of and that lie to the right of , while and are the parts of and that lie to the left of . The pointed vertex is the endpoint of . Proposition 5 in  guarantees that, if in the Tamari order, then and .
3. One can keep track of several parameters in the construction of Proposition 7. For instance, the number of non-initial contacts of is
3.2. From the construction to the functional equation
We now prove Proposition 5 through a sequence of lemmas. The first one describes in terms of homogeneous symmetric functions rather than power sums.
Let be defined by (12), and set
Hence is the homogenous symmetric function if is the power sum. Then the refined Frobenius series , defined by (7), can also be written as the following ordinary generating function:
where the sum runs over unlabelled -Tamari intervals , and is the height of the ascent of the upper path . In particular, is the ordinary generating function of -Tamari intervals, counted by the size (), the number of contacts (), and the distribution of ascents ( for each ascent of height in the upper path).
Let be an unlabelled Tamari interval, and let be the height of the ascent of . Denote . An -partitioned permutation is a permutation , together with a partition of the set of cycles of into labelled subsets , such that the sum of the lengths of the cycles of is . In the expression (7) of , the contribution of labelled intervals obtained by labelling in all possible ways is , where
In other words, is the exponential generating function of -partitioned permutations, counted by the size (variable ), the number of cycles in the block (variable ), and the number of cycles of length (variable ), for all . By elementary results on exponential generating functions, this series factors over ascents of . The contribution of the ascent is the exponential generating function of permutations of , counted by the size, the number of cycles of length for all , and also by the number of cycles if . But this is exactly (or if ), since
and the proof is complete111 An analogous result was used without proof in the study of the parking representation of the symmetric group [24, p. 28]..
We constantly use in this proof the inclusion given by Proposition 6. That is, we identify elements of with -Dyck paths having blocks. The size of an interval is thus now the number of blocks, and the height of the first ascent becomes the number of blocks in the first ascent.
Lemma 9 relies on the recursive description of Tamari intervals given in Proposition 7. We actually apply this construction to a slight generalization of -Tamari intervals. For , an -augmented -Dyck path is a Dyck path of size for some integer , where the first steps are up steps, and all the other up steps can be partitioned into blocks of consecutive up steps. The first steps of are not considered to be part of a block, even if is a multiple of . We denote by the number of blocks contained in the first ascent222Since the number of blocks does not depend on only, but also on , it should in principle be denoted . We hope that our choice of a lighter notation will not cause any confusion. of . A Tamari interval is an -augmented -Tamari interval if both and are -augmented -Dyck paths.
For let be the generating function of -augmented -Tamari intervals such that , counted by the number of blocks (variable ), the number of non-initial contacts (variable ) and the number of non-initial ascents of having blocks (one variable for each , as before). We are going to prove that for all ,
We claim that (16) implies Lemma 9. Indeed, -Tamari intervals coincide with -augmented -Tamari intervals. Since the initial contact and the first ascent are not counted in , but are counted in , the contribution in of intervals such that is , as stated in the lemma.
We now prove (16), by lexicographic induction on . For , the unique interval involved in is , where is the path of length . Its contribution is (since the initial and only contact is not counted). Therefore and (16) holds. Let and assume that (16) holds for all lexicographically smaller pairs . We are going to show that (16) holds at rank .
If and , then we are considering -augmented -Tamari intervals, that is, usual -Tamari intervals. But an -Tamari interval having blocks and blocks in the first ascent can be seen as an -augmented -Tamari interval having blocks and blocks in the first ascent, by considering that the first up steps of and are not part of a block. This implies that:
Now assume . The series counts -augmented -Tamari intervals such that . By Proposition 7, such an interval can be decomposed into a pointed interval and an interval (the “” in the notation is a bit unfortunate here; we hope it will not raise any difficulty). Note that is an -Tamari interval, while is an -augmented pointed -Tamari interval. Conversely, starting from such a pair , the construction of Proposition 7 produces an -augmented -Tamari interval, unless is not proper and . Moreover, . Using (14), we obtain:
where (resp. ) is the generating function of proper (resp. non-proper) pointed -augmented -Tamari intervals such that , counted by the number of blocks (variable ), the number of non-initial ascents of of height (variable ) for each , and the number of non-initial contacts of (variable ). The factor in (17) is the contribution of the interval .
To determine the series , expand the series as
where is the generating function of -augmented -Tamari intervals such that , and having non-initial contacts. Each such interval can be pointed in different ways to give rise to different proper pointed intervals , having respectively non-initial contacts. Therefore,
by the induction hypothesis. This is (16) at rank .
It remains to treat the case . To this end we need to determine the series . By definition, a pointed interval is non-proper if is empty, in which case can be identified with the (non-pointed) interval . This implies that . Therefore (17) and (18) give:
By the induction hypothesis, , so that
We recognise (16) at rank , and this settles the last case of the induction.