The Pillowcase Distribution and Near-Involutions

In the context of the Eskin-Okounkov approach to the calculation of the volumes of the different strata of the moduli space of quadratic differentials, the important ingredients are the pillowcase weight probability distribution on the space of Young diagrams, and the asymptotic study of characters of permutations that near-involutions. In this paper we present various new results for these objects. Our results give light to unforeseen difficulties in the general solution to the problem, and they simplify some of the previous proofs.


Introduction
A quadratic differential on a Riemann surface S is a function defined on the tangent bundle ω : T S → C that locally looks like ω(z) = f (z)(dz) 2 , for some meromorphic function f with at most simple poles. There is an interesting correspondence between these objects and flat surfaces that makes them useful in the study of many dynamical systems, notably billiards; see [26] for a good introduction to the subject.
We will say that two quadratic differentials ω and η defined on surfaces S and T are equivalent if there is a holomorphic diffeomorphism ϕ : S → T such that ϕ * η = ω. The set of equivalence classes is known as the moduli space. Its strata Mν are the subsets corresponding to quadratic differentials with fixed numbers of zeros and poles. We encode this information in a partition ν = (ν1 ≥ ν2 ≥ · · · ≥ ν k ≥ 1), in which a simple pole will be represented by a part νi equal to 1, a marked point will be a 2, and a zero of degree n will correspond to a part νi = n − 2. By Riemann-Roch, the genus g of the underlying Riemann surface is determined by |ν| − 2ℓ(ν) = zeros − poles = 4g − 4, where |ν| = i νi and ℓ(ν) is the number k of parts in ν.
The strata Mν of the moduli space are known to be complex manifolds of dimension 2g − 1 + ℓ(ν) or 2g − 2 + ℓ(ν), depending on whether a global square-root of the quadratic differential exists or not. Since a scalar multiple of a quadratic differential is another quadratic differential with the same number of zeros and poles, the strata are cones.
Local coordinates in Mν were constructed in [11]. The trick is to pass to the 2-fold coverS where an Abelian differential η (locally η = f (z) dz) exists that is a global square-root of the quadratic differential in question, ω = η 2 , and to take as coordinates the periods of η. The periods are defined to be the values of integrals of η along curves representing the generators of the homology group H1(S, P ; Z) of the surfaceS relative to the set P of zeros, poles, and marked points. Using these coordinates, one can pull Lebesgue measure from R dim Mν to Mν to define a volume element.
Using those same periods and the Riemann bilinear relations, a quadratic differential is seen to induce the area of the surface. One can thus take the set M 1 ν of quadratic differentials in Mν that induce area ≤ 1. This subset was shown to have finite volume [15,23].
The problem of the computation of the volume of different strata of the moduli space of quadratic differentials, that is, of the volume of M 1 ν , gained relevance as connections with Siegel-Veech constants and Lyapunov exponents of the geodesic flow in Teichmüller space were discovered; see for example [7,26].
The method developed by Eskin and Okounkov [6] gives a way to compute the volumes. Their method leveraged the correspondence of quadratic differentials with flat surfaces to construct a lattice F inside Mν consisting of tiled surfaces. These tiled surfaces are precisely the connected branched coverings of the pillowcase orbifold P = (R 2 /Z 2 )/{±1} with branching data ν ∪ (2, 2, . . . ) on one of the conic points of the pillowcase and (2, 2, . . . ) on the three others. (A quadratic differential on these coverings is obtained by pulling back the differential dz 2 defined on the pillowcase.) They proved that vol M 1 ν = lim d→∞ # of connected degree d coverings of P in Mν d dim Mν −1 .
They formed the generating function Zν of possibly disconnected coverings of the pillowcase in Mν graded by degree and weighted by where Aut(S) is the automorphism group of the covering S. (The weighting becomes asymptotically negligible as the degree increases, since most coverings have no nontrivial automorphisms; see [8].) This generating function is related to the generating function of connected coverings essentially by the exponential function; see [25,Chapter 3]. They thus showed that in order to find the volume, one needs to understand the leading terms of the asymptotics as q → 1, |q| < 1, of the series and that this series is a quasimodular form. The equivalence of (1) and (2) is classical and is explained for example in [18], [20,Section 1.3.7]. Here, • ν is a partition that indexes the stratum of the moduli space: the orders of the zeros of the quadratic differentials in the stratum are encoded as νi − 2 (νi = 2 correspond to marked points, and we do allow simple poles corresponding to the parts νi = 1), • the sum is taken over all partitions λ = (λ1, λ2, . . . ) of even integers |λ| = i λi, • dim λ denotes the dimension of the representation of the symmetric group corresponding to the partition λ, • (2, 2, . . . ) means the partition with |λ|/2 parts equal to 2, and similarly (ν, 2, 2, . . . ) means the partition that contains all the parts of ν and is completed with twos to be of the same size as λ, • the numbers fη(λ) = |Cη| χ λ (η) dim λ are the central characters of the symmetric group algebra, • |Cη| is the size of the conjugacy class of the symmetric group corresponding to permutations whose cycle type is given by the partition η, and • χ λ (η) is the character of one such permutation in the irreducible representation of the symmetric group corresponding to the partition λ.
In practical terms, this means that in order to determine the volume of one of these strata one needs to compute the first few coefficients of the series (2), from there deduce what polynomial in the Eisenstein series it corresponds to, and make use of the quasimodularity to determine the leading term in the asymptotics. This is involved because of the combinatoric calculation required in the first step, and it makes it impossible to conclude anything theoretically; it would be better to have a formula for the volumes. Our results go in this direction.
The alternative approach of Kontsevich [11,13] and Kontsevich and Zorich [12] has been expanded and fruitfully exploited by Athreya, Eskin, and Zorich [2,3], who have found exact formulas for the volumes of several families of strata.
Let us state our results. The form of the series (2) inspires the following definition [6]: Definition 1 (Pillowcase weight). For a partition λ, its pillowcase weight is defined by The pillowcase weights induce a probability distribution in the space of Young diagrams: we introduce a parameter q ∈ C, |q| < 1, and we let the weight of the partition λ be where Z = µ q |µ|/2 w(µ) is a normalization constant (independent of ν). Denoting the corresponding expectation by · w,q , we can rewrite the series (2) as where The goal of this paper is to analyze the probability distribution (3) and to give some results about the computation of the expectations of gν . With the following definition, we characterize the support of w.
Definition 2 (Balanced partition). A partition is balanced if either of the following equivalent statements is true: • the Young diagram of the partition λ can be constructed by adjoining 2-dominoes , • the 2-core of λ is empty, Our first result is an interesting formula, proved in Section 2: Using this formula, we obtain -through a variational argument-the following estimate; see Section 3 for the proof.
Proposition 4. Let ε > 0, and let Sε,n be the set of balanced partitions λ of n whose 2-quotients (α, β) satisfy Lα − L β > ε, where · denotes a Sobolev norm and Lα and L β are the rescaled contours of the partitions α and β (see Definitions 12 and 13). Then the w-probability of Sε,n is asymptotically of order O e −K √ n as n → ∞, for some K > 0 that depends on ε. In fact K → 0 as ε ց 0.
Whence we conclude that the measure is concentrated on the set of partitions that are close to having identical 2-quotient components α and β, that is, close to being constructed solely by 2 × 2 blocks: , and that it must be nearly uniform there. We also expect this to be true since, for large partitions, the odd hook lengths should almost cancel out the even hook lengths in the formula of Proposition 3.
Although we are unable to extract the uniformity from the result in Proposition 4, we can still prove it through the analysis of the expression for the n-point function obtained in [6]. Hence we have the following result, proved in Section 4.
Proposition 5. The probability distribution induced by the weights w(λ) induces a limit shape that coincides with the one for the uniform probability distribution, namely, it is the curve e −πx/ √ 6 + e −πy/ √ 6 = 1.
We were eager to find this result, because it determines immediately the asymptotic value of the expectations of the shifted power functions, and the function gν turns out to be a polynomial in the shifted power functions [6]. From there, we expected to find a Wick-type theorem that would further simplify our computation of the volumes. Instead we found (see Section 5 for details and for the proof): Proposition 6. The convergence to the limit shape is not normal. In other words, there is no central limit theorem for the pillowcase weights, and there is no Wick-type theorem associated to the distribution they induce.
We also find the following structural formula, which will be proved in Section 6 and commented in Remarks 33 and 34.
where • the sum is taken over all balanced partitions µ (see Definition 2) of size |µ| = |ν| whose Young diagram is completely contained 1 inside the Young diagram of λ, • oµ is the number of odd parts µi of µ, • sµ are the shifted Schur functions introduced in Definition 29 below, • z(ν) = ∞ n=1 n mn mn! (where mn is the number of parts νi equal to n) is the cardinality of the centralizer of an element of cycle type ν in the symmetric group, and • the pairs of partitions (α, β) and (a, b) are the 2-quotients of λ and µ, respectively.
In the final Section 7, we discuss the initial applications of these results to the calculation the expectations (4).
This paper elaborates on some of the results of the author's PhD thesis [20], where a more detailed account of most of the proofs can be found.
Acknowledgements The author is deeply indebted to his doctoral adviser Professor Andrei Okounkov, who suggested the problem and provided much valuable insight in the course of numerous conversations, and to Professor Grigori Olshanski for several useful discussions and for doing a full reading of the thesis this paper is based on. The author is also grateful for the suggestions of the editors of the journal. The author is also grateful for the support of conacyt-Mexico, the Mexican Secretariat of Public Education, and Princeton University.

Formula in terms of hooks
Proof of Proposition 3. Let λ be a balanced partition. Note first that and recall the classical result that [6], [20, Section 2.2.1] where the pair of partitions (α, β) is the 2-quotient of λ. Since λ is balanced, exactly half of its hook lengths are even, and the hook lengths of α and β are in one-to-one correspondence with the even hook lengths of λ divided by 2. (See [20, Section 2.1].) Using this equality and the hook formula of Frame-Robinson-Thrall, namely, where h denotes the length of the hook of the cell ∈ λ, we get where the product is taken over those cells whose hook lengths are even.
In the denominator we have the product of the halves of the even hook lengths of λ. In other words, Since hook lengths of λ ( even hook lengths of λ) 2 = odd hook lengths of λ even hook lengths of λ , plugging (6), (7) and (8) into the expression given in Definition 1 for w(λ), we get the desired formula.

Concentration near the diagonal
The goal of this section is to prove Proposition 4. We first collect some definitions and basic results.
Definition 8 (Rescaled rim function). Let λ be a partition of n = |λ|. We contract its diagram until it has area 1, rescaling by 1/ √ n, and we associate to it the rescaled rim function of λ, namely, the non-increasing function F : R+ → R+ given by Definition 9 (Approximate hook). For an increasing function F : its approximate hook at (x, y).
Lemma 10 (Kerov-Vershik [24]). Let λ be a partition and let ∈ λ be a cell in its Young diagram, and denote by F its rescaled rim function.
Proof of Lemma 11. We have Since c is decreasing and c(h) → ∞ as h → ∞, the worst case scenario is the case in which we maximize the number of small odd hook lengths h . This happens in the case of the staircase partition (ℓ, ℓ − 1, . . . , 2, 1), and in this case the right hand side of (9) is of order O |λ| .
Definition 12 (Rescaled contour of a partition). The rescaled contour of the Young diagram of a partition λ is the function L λ : R → R+ whose graph is the union of the graphs of the rescaled rim function F of λ and of F −1 , after the change of variables In other words, we rotate the rescaled diagram of λ so that its sides become aligned with the graph of x → |x|; then L λ is the function that describes the rim of the rotated diagram, and it equals |x| outside the diagram.

Proposition 14.
As |λ| → ∞ (and restricting to λ balanced), where ∆ = Lα − L β is the difference of the rescaled contours of the components α and β of the 2-quotient of λ.
Proof of Proposition 14. Consider the diagram of a balanced partition λ rescaled by 1/ √ n. Denote by O and E the domains inside this diagram corresponding to the cells whose hooks are of odd and even length, respectively. It follows from Lemma 10 that In a balanced partition, the area of O is the same as the area of E, so the contributions of log √ n cancel out. So √ n can be removed from the integrand, and from Lemma 11 we have We change variables to s and t such that , and the first term above becomes whereÕ andẼ are the images of O and E under the change of variables. Let (α, β) be a pair of partitions giving the 2-quotient of λ. Since the Maya diagram of λ is a sequence that can be obtained from the Maya diagrams of α and β by placing their elements alternatingly, we have Again referring to the Maya diagram of λ, note that a hook of length k corresponds to a pair of slots k units apart such that the slot on the left has a pebble and the slot on the right does not [21,Excercise 7.59]. In particular, it follows that the integral overẼ involves the interaction of each component of the 2 quotient with itself, so this term becomes Similarly, the integral overÕ involves interactions of the components of the 2-quotient with each other, and becomes Note the implicit change of coordinates here, which would handle the translations by k/ 2 √ 2n in formula (10). Adding (11) and (12), and simplifying, we obtain where ∆(s) = Lα(s) − L β (s), from where the statement of the proposition follows after integrating by parts twice.
Lemma 15. As n → ∞, we have the asymptotic behavior where the sum is taken over all λ of size |λ| = n.
Proof of Lemma 15. It was shown in Eskin-Okounkov [6] that We have thus the following parameters for the Theorem of Meinardus, as presented by Andrews [1,Chapter 6]: q in the book is q 2 here, an = 1 2 , The statement of the lemma follows.
Proof of Proposition 4. We can bound the cardinality of Sε,n by the total number p(n) of partitions of n, for which the asymptotics is well known [1, Theorem 6.3]. The w probability of λ is given by The statement of the proposition follows from Proposition 14, Lemma 15, and the asymptotic relation (13).

The limit shape
Our goal in this section is to prove Proposition 5. We first need a few results and definitions.
Recall that the rescaled contour L λ of λ was introduced in Definition 12.
Definition 16 (Limit shape). Let ν be a measure on the set of Young diagrams, such that ν({λ : |λ| = n}) = 1 for all n > 0. A continuous function Ω : R → R+ such that R [Ω(x) − |x|] dx = 1 is the limit shape induced by ν if there is some function e : R+ → R+ such that e(x) → 0 as x → 0 and for all ε > 0 there is some N ≫ 0 such that, for all n > N , Definition 17 (Moments of the limit shape). For a limit shape Ω, we define its moments Definition 18 (Shifted power functions). Let The shifted power function pµ indexed by the partition µ is defined as Remark 19. When a distribution ν on the set of Young diagrams has a limit shape, lim n→∞ p k ν,n = kµ k−1 where · ν,n denotes the mean among all partitions of size |λ| = n. (See for example [10], [20, Section 3.1].) Most of our derivation will come from Theorem 24 below. In order to state it, we need some more definitions.
Definition 20 (Jacobi theta function). The Jacobi theta function is given by Remark 21. Recall the definition [16] of the classical Jacobi theta functions θ jk , j, k = 0, 1: In terms of these and as a consequence of the Jacobi triple product formula, where x = e u , q = e −h , and η stands for the Dedekind eta function, Definition 22. The n-point function is the generating function Remark 23. Since [5,6] we have u i 1 1 · · · u in n i1! · · · in! pi 1 · · · pi n w,q + irrelevant terms.
(The terms missing in this formula are irrelevant in the sense that they do not encode information of interest to us. They correspond to monomials in the variables ui that are different from the ones in the sum.) Hence, in view of Remark 19, a statement about F should be interpreted as a statement about the moments of the limit shape.
Theorem 24 (Eskin-Okounkov [6, Theorem 5]). We have where the brackets [·] indicate the operation of taking the coefficient of the indicated monomial, and the series expansion is performed in the domain as h → +0.
For the proof of Proposition 25 we need two lemmas. We will use the notation ≈ to mean "up to exponentially small terms."

Lemma 26 ( [8, Proposition 4.1]). We have
as h → +0, uniformly in u. This asymptotic relation can be differentiated any number of times.
Lemma 27. We have the following approximations for ϑ at e u = 1 and e u = −1, respectively: stands for the integer closest to x, and Proof of Lemma 27. This is a straightforward application of the modular transformation h → −1/h in the expression (14). We also use the identity ϑ11(−e u , q) = −ϑ10(e u , q).
Proof of Proposition 25. We start with the expression from Theorem 24. We first need to understand the asymptotic behavior of the coefficient of y 0 1 · · · y 0 n . We will approach this as an integral where Q is the quotient of theta functions to the right of [y 0 1 · · · y 0 n ] in Theorem 24, including the part with the square root. The approximation of Lemma 27 implies that if xi = e u i h , yi = e v i , and q = e −h , then Q is the exponential of a sum of many terms. Some of them vanish as h → 0; for the rest, we have to take l'Hôpital's rule. In the end, we see that Q tends to We take the integral for each vi on the segment [0, 2πi] ⊂ C, and we get i π cos(πui).
Finally we apply the approximation of Lemma 26 to the product i 1/ϑ(xi) in front of the expression of Theorem 24.
Corollary 28. The expectations of the shifted power functions are asymptotically multiplicative, namely Proof of Corollary 28. This follows from Proposition 25 and from the estimate [6, Section 3.3.5] where ℓ(ρ) denotes the number of parts of the partition ρ.
Proof of Proposition 5. In order to determine that the limit shape should be equal to the one induced by the uniform distribution, we will show that, in the q → 1 limit, the asymptotic behavior of the expectations p k w,q coincides with µ unif k−1 /k, where µ unif k are the moments of that limit shape (compare with Remark 19). Because these moments do not grow very fast, they determine the shape uniquely. A sufficient condition [4,Chapter 30] for this to be the case is that lim sup which is true in this instance.
In the case of the q-coupled uniform distribution, in which the weight of the partition λ is q |λ| , the corresponding n-point function (which is also the exponential generating function of the numbers p k unif,q ) is where x = e u .The proof is similar to that of Theorem 24, noting that (in the notations of [6]) F unif (x) = 1 λ q |λ| [y 0 ] tr q H ψ(xy)ψ * (y).

Asymptotically, this behaves as
where q = e −h , h → +0, and ≈ means "up to exponentially small terms." These asymptotics are obtained through an application of Lemma 26. From Proposition 25 we see that the 1-point function F for the pillowcase distribution is (up to a rescaling) asymptotically equivalent to F unif , whence the moments of the limit shape are also the same, µj = µ unif j .
On the other hand, from the multiplicativity property of Corollary 28 it follows that for any partition ρ, the variance Var pρ = p 2 ρ w,q − pρ 2 w,q → 0 very quickly as q → 1. This means that the probability must be concentrated at one point. In other words, our candidate is indeed the limit shape.

No central limit theorem
Proof of Proposition 6. If the probability measure were asymptotically gaussian, a Wick-type theorem would hold for the shifted power functions pµ. Namely, we would have an identity of the type abcd w,q = ab w,q cd w,q + ac w,q bd w,q + ad w,q bc w,q for all functions a, b, c, and d with vanishing (w, q)-mean and contained in the algebra generated by the functions p k . In particular, this means that we would need to have and To check whether this was the case, we computed the first few terms of the following series, and then we used the quasimodularity property proved by Eskin-Okounkov [6, Theorem 1] with the methodology detailed in Appendix A to get: Here, "e.s.t." stands for exponentially small terms.
Instead of (15), we get Instead of (16), we get We conclude that the convergence is not asymptotically gaussian and that no Wick-type theorem applies to this distribution. which allows us to take inverse limits, just as in de definition of symmetric functions (see for example [14]). The resulting objects, sµ(x1, x2, . . . ), are known as shifted Schur functions [17] (or Frobenius-Schur functions [19]).

Characters of near-involutions
Proposition 30. For λ and ν balanced partitions, with the Young diagram of ν entirely contained inside the Young diagram of λ, where the sum is taken over all balanced partitions µ with |µ| = |ν| and whose Young diagrams are completely contained in the diagram of λ, and the pairs of partitions (α, β) and (a, b) are the 2-quotients of λ and µ, respectively.
Remark 31. This, together with the formula from the following lemma (in the case of µ empty) and together with any of the available expressions for the dimension dim λ of a partition (such as the Frame-Robinson-Thrall hook formula), gives an explicit formula for χ λ (ν, 2, 2, . . . , 2). Proof of Lemma 32. The is a straightforward consequence of the definition of the 2-quotients and of the Murnaghan-Nakayama rule; see [20] for details.
Proof of Proposition 30. First, observe that the Murnaghan-Nakayama rule implies that The sum is over all partitions µ of size |ν| whose diagram is completely contained inside the diagram of λ. Clearly, χ λ/µ (2, 2, . . . , 2) vanishes unless µ is balanced, so all sums from this point on will be over balanced partitions µ of size |ν| and contained in λ. To this expression, we apply the formula from Lemma 32, and we apply the same formula with empty µ to the denominator, to get where o is the number of odd parts in µ; α, β, a, and b are as in the statement of the proposition, and Now we apply, to each of the quotients of dimensions, the Okounkov-Olshanski formula [17,Equation 0.14] where n = |λ| and k = |µ|. We get exactly the formula in the proposition because |λ| 2 − |α| = |β| and |λ/µ| 2 − |α/a| = |β/b|.

Moduli spaces
The function appears in the work of Eskin-Okounkov [6] as an essential ingredient for the computation of the volumes of the different strata of the moduli space of quadratic differentials. The partition ν determines the corresponding stratum: the multiplicities of the zeros of the quadratic differentials are encoded as νi −2. Simple poles are allowed and correspond to parts νi = 1. It was shown in [6] that these volumes are equal to the first term of the asymptotics of the expectations gν w,q as q → 1. See also [20] for a detailed account. Remark 33. In Eskin-Okounkov [6], it is proved that these expectations are quasimodular forms. Using Proposition 7, we break up these expectations into linear combinations of expectations of the form Using methods similar to those of [6, Section 3], it can be proven [20, Section 2.5] that these are also quasimodular when taken individually. This arguably shortens the proof of the quasimodularity of Zν (q) since most of Section 2 of [6] becomes superfluous.
In other words, the matrix relating Vν to Hµ is the transpose of the inverse, M −T . Whence we also know that (asymptotically) In this way, Proposition 7 reveals some structure in the problem of determining gν w,q in general.
We shall now derive some results about the asymptotic analysis of gν w,q .
Let C1 be the space of functions r : R → R, r(x) ≥ |x|, with Lipschitz constant ≤ 1, that is, for all x, y ∈ R, Note that all contours L λ of partitions λ belong to C1, as do all possible limit shapes. We endow C1 with the topology of the supremum norm.
For a function f : C1 → R and a partition λ, we define f (λ) to be the function evaluated on the contour of the partition, f (L λ ).
Corollary 35. Let f, g : C1 → R be two functions with finite w-expectations at each level n. Assume that |f (λ)| and |g(λ)| increase at most polynomially as |λ| → ∞. In particular for g, for some b > 0. Assume additionally that g(λ)|λ| −b is continuous with the topology of C1. Then where (α, β) is the 2-quotient of the partition λ over which the sum of the expectation is taken.
Proof of Corollary 35. Let ε > 0. Let δ > 0 be such that if Lα −L β < δ then |g(Lα) − g(L β )| < εn b . We want to show that the following tends to 0: Here, the summation can be split into two parts, The later can be bounded easily using Proposition 4: we get for some constant K δ that depends on δ. Since f and g grow polynomially, this tends to 0 as n → ∞. On the other hand, the first part in the summation above goes like Remark 36. It follows from Corollary 35 and Proposition 5 that the components α and β of the 2-quotient of λ also have a limit shape, which in fact coincides with the one corresponding to the uniform distribution.
Proposition 37. With the same notations as above + terms of lower degree, where K > 0, both sums are taken over balanced partitions µ of size |µ| = |ν| and Moreover, the first term (18) of gν w always vanishes.

A Program listings
In this section we present a program that we have used to compute the asymptotic behavior of the expectations used in Section 5 for the proof of Proposition 6. The program is a script designed to run in Sage [22]. What the program does is essentially a linear regression. We know from the work of Okounkov and Eskin [6] that f w,q is a quasimodular form. Here, f is the desired function and it must be in the algebra generated by the shifted power functions pn; f appears in the program below with the name function of interest. We compute the first few coefficients of this series, as well as the first few elements of a basis of the space of quasimodular forms, and then we invert the corresponding matrix to find out what the linear combination is. Our basis is given by products of the Eisenstein series E2(q 2 ), E2(q 4 ), E4(q 4 ), and it is hence very easy to compute the asymptotic behavior of each term.
The hardest part is the computation of the coefficients of the series Z f w,q because each of them is a combinatorial sum over all pairs of partitions whose sizes add up to the corresponding power of q. This is done by the slave program, which we present in Section A.1. The computation of the basis and the linear regression are done by the main program, presented in Section A.2.
The output of the program is a polynomial in the variable h. In the notations of Section 5, h is really a placeholder for h −1 . This polynomial thus respresents the non-exponentially small part the asymptotic behavior of f w,q . If too few coefficients of f w,q have been computed with the slave program, the main program will effectively return a lot of trash.

A.1 Slave program
The following function computes very quickly the value of w(λ), and it takes three partitions: p equal to λ, and also p1 and p2 equal to α and β, the components of the 2-quotient of λ.