Counting quadrant walks via Tutte’s invariant method
July 9, 2019
In the 1970s, William Tutte developed a clever algebraic approach, based on certain “invariants”, to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past 15 years to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps, taken in .
We first adapt Tutte’s approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic. This includes Gessel’s famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity follows almost automatically.
Then, we move to a complex analytic viewpoint that has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is D-algebraic (that is, satisfies polynomial differential equations).
Key words and phrases:Lattice walks, enumeration, differentially algebraic series, conformal mappings
A tribute to William Tutte (1917-2002), on the occasion of his centenary
In the past 15 years, the enumeration of plane walks confined to the first quadrant of the plane (Figure 1) has received a lot of attention, and given rise to many interesting methods and results. Given a set of steps and a starting point (usually ), the main question is to determine the generating function
where is the number of -step quadrant walks from to , taking their steps in . This is one instance of a more general question consisting in counting walks confined to a given cone. This is a natural and versatile problem, rich of many applications in algebraic combinatorics [BM11, CDD07, GZ92, Gra02, Kra89], queuing theory [AWZ93, CB83, FH84], and of course in enumerative combinatorics via encodings of numerous discrete objects (e.g. permutations, maps…) by lattice walks [Ber07, BGRR, GWW98, KMSW15, LSW17].
At the crossroads of several mathematical fields. Most of the recent progress on this topic deals with quadrant walks with small steps (that is, ). Then there are inherently different and relevant step sets (also called models) and a lot is known on the associated generating functions . One of the charms of these results is that their proofs involve an attractive variety of mathematical fields. Let us illustrate this by two results:
a certain group of birational transformations associated with the model plays a crucial role in the nature of . Indeed, this series is D-finite (that is, satisfies three linear differential equations, one in , one in , one in , with polynomial coefficients in and ) if and only if is finite. This happens for 23 of the 79 models. The positive side of this result (D-finite cases) mostly involves algebra on formal power series [BM02, BMM10, Ges86, GZ92, Mis09]. The negative part relies on a detour via complex analysis and a Riemann-Hilbert-Carleman boundary value problem [Ras12, KR12], or, alternatively, on a combination of ingredients coming from probability theory and from the arithmetic properties of G-functions [BRS14]. The complex analytic approach also provides integral expressions of in terms of Weierstrass’ function and its reciprocal.
Among the 23 models with a D-finite generating function, exactly 4 are in fact algebraic over (that is, satisfies a polynomial equation with polynomial coefficients in , and ). For the most mysterious of them, called Gessel’s model (Fig. 1, left), a simple conjecture appeared around 2000 for the numbers , but resisted many attempts during a decade. A first proof was then found, based on subtle (and heavy) computer algebra [KKZ09]. The algebraicity was only discovered a bit later, using even heavier computer algebra [BK10]. Since then, two other proofs have been given: one is based on complex analysis [BKR17], and the other is, at last, elementary [BM16a].
Classifying solutions of functional equations. Beyond the solution of a whole range of combinatorial problems, the enumeration of quadrant walks is motivated by an intrinsic interest in the class of functional equations that govern the series . These equations involve divided differences (or discrete derivatives) in two variables. For instance, for Kreweras’ walks (steps , , ), there holds:
This equation is almost self-explanatory, each term corresponding to one of the three allowed steps. For instance, the term
counts walks ending with a West step, which can never be added at the end of a walk ending on the -axis. The variables and are sometimes called catalytic. Such equations (sometimes linear as above, sometimes polynomial) occur in many enumeration problems, because divided differences like
have a natural combinatorial interpretation for any generating function . Examples can be found in the enumeration of lattice paths [BF02, BMM10, BMP00], maps [Bro65, BT64, CF16, Tut95], permutations [BM11, BM03, BGRR]… A complete bibliography would include hundreds of references.
Given a class of functional equations, a natural question is to decide if (and where) their solutions fit in a classical hierarchy of power series:
where we say that a series (say in our case) is D-algebraic if it satisfies three polynomial differential equations (one in , one in , one in ). An historical example is Hölder’s proof that the Gamma function is hypertranscendental (that is, not D-algebraic), based on the difference equation . Later, differential Galois theory was developed (by Picard, Vessiot, then Kolchin) to study algebraic relations between D-finite functions [vdPS03]. This theory was then adapted to -equations, to difference equations [vdPS97], and also extended to D-algebraic functions [Mal04]. Let us also cite [DHRar] for recent results on the hypertranscendence of solutions of Mahler’s equations.
Returning to equations with divided differences, it is known that those involving only one catalytic variable (arising for instance when counting walks in half-plane) have algebraic solutions, and this result is effective [BMJ06]. Algebraicity also follows from a deep theorem in Artin’s approximation theory [Pop86, Swa98]. Moreover, as described earlier in this introduction, the classification of quadrant equations (with two catalytic variables and ) with respect to the first three steps of the hierarchy (2) is completely understood. One outcome of this paper deals with the final step: D-algebraicity.
Contents of the paper. We introduce two new objects related to quadrant equations, called invariants and decoupling functions. Both are rational functions in and . Not all models admit invariants or decoupling functions. We show that these objects play a key role in the classification of quadrant walks:
first, we prove that invariants exist if and only if the group of the model is finite (that is, if and only if is D-finite; 23 cases). In this case, decoupling functions exist if and only if the so-called orbit sum vanishes (Section 4). This holds precisely for the algebraic models (Figure 2, left).
In those cases, we combine invariants and decoupling functions to give short and uniform proofs of algebraicity. This includes the shortest proof ever found for Gessel’s famously difficult model, and extends to models with weighted steps [KY15], for which algebraicity was sometimes still conjectural (Sections 3 and 4).
Models with an infinite group have no invariant. But we define for them a certain (complex analytic) weak invariant, which is explicit. Then for the infinite group models that admit decoupling functions, we give a new, integral free expression of (Section 5). This expression implies that is D-algebraic in , and . We compute explicit differential equations in for (Section 6).
Note that an extended abstract of this paper appeared in the proceedings of the FPSAC’16 conference (Formal power series and algebraic combinatorics [BBMR16]). Moreover, two recent preprints of Dreyfus et al. essentially complete the differential classification of quadrant walks by proving that the remaining 47 infinite group models are not D-algebraic in (nothing is known regarding the length variable ) [DHRS17, DHRSon]. The proof relies on Galois theory for difference equations. The nature of , for quadrant models with small steps, can thus be summarized as follows:
|Decoupling function||No decoupling function|
|Rational invariant ( Finite group)||Algebraic||D-finite transcendental|
|No rational invariant ( Infinite group)||Not D-finite, but||Not D-algebraic (in )|
Note that the existence of invariants only depends on the step set , but the existence of decoupling functions is also sensitive to the starting point: in Section 7, we describe for which points they actually exist. In particular, we show that some quadrant models that have no decoupling function (and are not D-algebraic) when starting at still admit decoupling functions when starting at other points. Even though we have not worked out the details, we believe them to be D-algebraic for these points.
This paper is inspired by a series of papers published by Tutte between 1973 and 1984 [Tut73, Tut84], and then surveyed in [Tut95], devoted to the following functional equation in two catalytic variables:
This equation appears naturally when counting planar triangulations coloured in colours. Tutte worked on it for a decade, and finally established that is D-algebraic in . One key step in his study was to prove that for certain (infinitely many) values of , the series is algebraic, using a pair of (irrational) series that he called invariants [Tut95]. They are replaced in our approach by (rational) invariants and decoupling functions. After an extension of Tutte’s approach to more general map problems [BBM11, BBM17], this is now the third time that his notion of invariants proves useful, and we believe it to have a strong potential in the study of equations with divided differences.
2. First steps to quadrant walks
We now introduce some basic tools in the study of quadrant walks with small steps (see e.g. [BMM10] or [Ras12]). A simple step-by-step construction of these walks gives the following functional equation:
is the kernel of the model. It is a polynomial of degree in and , which we often write as
We shall also denote
Note that , so that the basic functional equation (4) reads
Seen as a polynomial in , the kernel has two roots and , which are Laurent series in with coefficients in . If the series is well defined, setting in (6) shows that
If this holds for and , then
This equation will be crucial in our paper.
We define symmetrically the roots and of (when solved for ).
The group of the model, denoted by , acts rationaly on pairs , which will typically be algebraic functions of the variables , and . It is generated by the following two transformations:
where the polynomials and are the coefficients of defined by (5). Note that these transformations do not depend on , although does. Indeed,
and symmetrically for . Both transformations are involutions, thus is a dihedral group, which, depending on the step set , is finite or not. For instance, if , the basic transformations are
with and , and they generate a group of order 6:
where and . One key property of the transformations and is that they leave the step polynomial, namely
More generally, since , every element in the orbit of (or ) satisfies .
A step set is singular if each step satisfies .
The above constructions (functional equation, kernel, roots, group…) can be extended in a straightforward fashion to the case of weighted steps. In particular, the kernel becomes:
where is the weight of the step .
Notation. For a ring , we denote by (resp. , ) the ring of polynomials (resp. formal power series, Laurent series) in with coefficients in . If is a field, then stands for the field of rational functions in . This notation is generalized to several variables. For instance, the series belongs to . The valuation of a series in is the smallest such that the coefficient of is non-zero.
We will often use bars to denote reciprocals (as long as we remain in an algebraic, non-analytic context): , .
3. A new solution of Gessel’s model
This model, with steps , appears as the most difficult model with a finite group. Around 2000, Ira Gessel conjectured that the number of -step quadrant walks starting and ending at was
where is the rising factorial. This conjecture was proved in 2009 by Kauers, Koutschan and Zeilberger [KKZ09]. A year later, by a computer algebra tour de force, Bostan and Kauers [BK10] proved that the three-variate series is not only D-finite, but even algebraic. Two other more “human” proofs have then been given [BKR17, BM16a]. Here, we give yet another proof based on Tutte’s idea of invariants.
The basic functional equation (6) holds with , , and . It follows from that
In Tutte’s terminology, is a (rational) -invariant.
Let us take , where is a new variable and stands for . Then and are Laurent series in with coefficients in , satisfying
The series and simply differ by the transformation . For , the series and are well defined as series in (with coefficients in ).
The expansions of the near are found either by solving explicitly , or using Newton’s polygon method [Abh90]. To prove the second point, let us write
where is the number of quadrant walks consisting of North-East steps, East steps, South-West steps and West steps. Given that and are series in with respective valuation and , the valuation of the summand associated with the 4-tuple is
For to be well defined, we want that for any , only finitely many 4-tuples satisfy , and . The above expression of shows that , and must be bounded (for instance by ), and the inequality bounds as well. Hence is well defined.
This implies that is also well defined, as is just obtained by selecting in the expression of the 4-tuples such that .
so that we can rewrite (8) as
This should be compared to (11). In Tutte’s terminology, the series is, as , an invariant, but this time it is (most likely) irrational. The connection between and will stem from the following lemma, which states, roughly, that invariants with polynomial coefficients in are trivial.
Let be a Laurent series in with coefficients in , of the form
for some . For , the series and are well defined Laurent series in , with coefficients in . If they coincide, then is in fact independent of .
By considering , we can assume that . In this case,
Assume that is not uniformly zero, and let
(see Figure 4 for an illustration). The inequalities imply that is finite, at least equal to . Moreover, only finitely many pairs satisfy and .
Since , any series can be substituted for in , and
Applying this to and , and writing that , shows that
Hence the polynomial must vanish, which is incompatible with the definition of .
with . Still, we can construct from them a series satisfying the assumptions of the lemma. First, we eliminate the simple pole of at by considering , which still takes the same value at and . The coefficients of this series have a pole of order at most at . By subtracting an appropriate series of the form , where and depend on but not on , we obtain a Laurent series in satisfying the assumptions of the lemma: the polynomiality of the coefficients in holds by construction, and the fact that in each monomial , the exponent of is (roughly) at most half the exponent of comes from the fact that this holds in , due to the choice of the step set (a walk ending at has at least steps). Thus this series must be constant, equal for instance at its value at . In brief,
for some series , , , in . Expanding this identity near determines the series , , , in terms of , and gives the following equation.
Replacing in (15) the series and by their expressions (in terms of , and ) gives for a cubic equation, involving , , and three additional unknown series in , namely , and . It is not hard to see that this equation defines a unique 4-tuple of power series, with and in .
Equations of the form
occur in the enumeration of many combinatorial objects (lattice paths, maps, permutations…). The variable is often said to be a catalytic variable. Under certain hypotheses (which essentially say that they have a unique solution in the world of power series, and generally hold for combinatorially founded equations), their solutions are always algebraic, and a procedure for solving them is given in [BMJ06].
Applying the procedure of [BMJ06] to the equation obtained above for Gessel’s walks shows in particular that and belong to , where is the unique series in with constant term 1 satisfying . Details on the solution are given in Appendix A.4. Let us mention that, in the other “elementary” solution of this model, one has to solve an analogous equation satisfied by [BM16a, Sec. 3.4]. Once is proved to be algebraic, the algebraicity of , and finally of , follow using (7) and (6).
4. Extensions and obstructions: uniform algebraicity proofs
We now formalize the three main ingredients in the above solution of Gessel’s model: the rational invariant given by (11), the identity (12) expressing as a difference , and finally the “invariant Lemma” (Lemma 2). We discuss the existence of rational invariants , and of decoupling functions , for all quadrant models with small steps. In particular, we relate the existence of rational invariants to the finiteness of the group . Finally, we show that the above solution of Gessel’s model extends, in a uniform fashion, to all quadrant models (possibly weighted) known or conjectured to have an algebraic generating function (see Figure 2). They are precisely those that have a rational invariant and a decoupling function. For one of them, we need an algebraic variant of the invariant lemma, which is described in Section 4.4.
A quadrant model admits invariants if there exist rational functions and , not both in , such that as soon as the kernel vanishes. Equivalently, the numerator of , written as an irreducible fraction, contains a factor . The functions and are said to be an -invariant and a -invariant for the model, respectively.
Our definition is more restrictive than that of Tutte [Tut95], who was simply requiring and to be series in with rational coefficients in (or ).
Assume that there exists a rational function such that when and are the roots of the kernel , solved for . Then is a rational function of , and forms a pair of invariants.
We have , hence is a rational function of and as any symmetric function of the roots and . The property tells us that equals as soon as . which is the condition of Definition 4.
Example. In Gessel’s case, was given by (11), and we find
We can also check that divides . Indeed,
The factor shows that the pair forms a pair of invariants for the model obtained by reflection in a vertical line.
In fact, two models differing by a symmetry of the square have (or have not) invariants simultaneously. Since these symmetries are generated by the reflections in the main diagonal and in the vertical axis, it suffices to consider these two cases.
Take a model with kernel and its diagonal reflection , with kernel Then admits invariants if and only if does, and in this case a possible choice is and A similar statement holds for the vertical reflection , with kernel where . A possible choice is then and
The proof is elementary.
We can now tell exactly which models admit invariants. Note that it is easy to decide whether a given pair is a pair of invariants: it suffices to check whether has a factor . But the following result tells us how to construct such pairs.
A (possibly weighted) quadrant model has rational invariants if and only if the associated group defined by (9) is finite.
Assume this is the case, and let be a rational function in . Consider the rational function
are respectively rational functions in and , and they form a pair of invariants, as long as they do not both belong to .
Assume that the model has invariants , with . If is any element in the orbit of , then , hence . But the form of and implies by transitivity that and .
Assume that the group of the model is infinite. Then the orbit of is infinite as well. Indeed, if it were finite, then there would exist , different from the identity, such that . Denoting , where both coordinates and are in , this would mean in particular that vanishes at , forcing this rational function to zero, or to have a factor in its numerator. But this is impossible since does not involve the variable (while does), hence . By the same argument, , hence is the identity, which contradicts our assumption. Hence the orbit of is infinite. This implies that infinitely many series occur in it (as the first coordinate of a pair), and thus the equation (in ) has infinitely many solutions. This is clearly impossible since we have assumed that .
Conversely, take a model with finite group, a rational function in , and define as above. For instance, for a model with a vertical symmetry, has order 4, and the orbit of reads:
Thus if we take , then and .
Returning to a general group, observe that takes the same value, by construction, on all elements of the orbit of . In particular, . Hence the above defined series and are rational in and (or and ). Moreover, , being the sum of over the orbit of , coincides with , and is a pair of invariants (unless and both depend on only).
For instance, for the reverse Kreweras model , and , we find . But taking instead gives true invariants:
Let us finally prove that, for any weighted model, there exists such that the invariant obtained from the function actually depends on . Assume this is not the case. Let have order , and let be the distinct series that occur in the orbit of as the first coordinate of some pair. Then by assumption, is an element of for all , which shows that all symmetric functions of the ’s depend on only. This implies that each is an algebraic function of only, which is impossible since .
Since we mostly focus in this section on algebraic quadrant models, we have only given in Table 1 explicit invariants for the four algebraic (unweighted) models. The remaining models with a finite group either have a vertical symmetry (in which case they admit as -invariant), or differ from an algebraic model by a symmetry of the square (in which case Lemma 6 applies). Invariants for the four weighted models of Figure 2 are given in Table 3.
|see Lemma 6|
In Section 5, we introduce a weaker notion of (possibly irrational) invariants, which guarantees that any non-singular quadrant model now has a weak invariant. One key difference with the algebraic setting of this section is that the new notion is analytic in nature.
4.2. Decoupling functions
We now return to the identity (12), which we first formalize into an apparently more demanding condition.
A quadrant model is decoupled if there exist rational functions and such that as soon as . Equivalently, the numerator of , written as an irreducible fraction, contains a factor . The functions and are said to form a decoupling pair for the model.
Again, this is equivalent to a statement involving a single function , as used in the previous section (see (12)).
Assume that there exists a rational function such that
where and are the roots of the kernel , solved for . Define . Then , and is a decoupling pair for the model.
hence is a rational function of and since it is symmetric in and . The property tells us precisely that the condition of Definition 8 holds.
Example. In Gessel’s case, we had (see (12)), corresponding to .
with . In Tutte’s terminology, this would make a second “invariant”. But our terminology is more restrictive, as our invariants must be rational.
Now, which of the quadrant models are decoupled? Not all, at any rate: for any model that has a vertical symmetry, the series are symmetric in and , and so any expression of of the form would be at the same time an expression of .
In the case of a finite group, we give in Theorem 12 below a criterion for the existence of a decoupling pair, as well as an explicit pair when the criterion holds. This shows that exactly four of the finite group models are decoupled (and these are, as one can expect from the algebraicity result of Section 3, those with an algebraic generating function). The four weighted models of Figure 2, right, are also decoupled.
For models with an infinite group, we have first resorted to an experimental approach to construct invariants. Indeed, one can look for decoupling functions by prescribing the form of the partial fraction expansion of : we first set