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Towards the Distribution of the Size of a Largest Planar Matching

and Largest Planar Subgraph in Random Bipartite Graphs

MARCOS KIWI∗

Depto. Ing. Matem´ atica and

Ctr. Modelamiento Matem´ atico UMI 2807,

University of Chile

Correo 3, Santiago 170–3, Chile

e-mail: mkiwi@dim.uchile.cl

MARTIN LOEBL†

Dept. of Applied Mathematics and

Institute of Theoretical Computer Science (ITI)

Charles University

Malostransk´ e n´ am. 25, 118 00 Praha 1

Czech Republic

e-mail: loebl@kam.mff.cuni.cz

Abstract

We address the following question: When a randomly chosen regular bipartite multi–graph is drawn in the plane

in the “standard way”, what is the distribution of its maximum size planar matching (set of non–crossing disjoint

edges) and maximum size planar subgraph (set of non–crossing edges which may share endpoints)? The problem

is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam’s problem). We present

combinatorial identities which relate the number of r-regular bipartite multi–graphs with maximum planar matching

(maximum planar subgraph) of at most d edges to a signed sum of restricted lattice walks in Zd, and to the number

of pairs of standard Young tableaux of the same shape and with a “descend–type” property. Our results are obtained

via generalizations of two combinatorial proofs through which Gessel’s identity can be obtained (an identity that is

crucial in the derivation of a bivariate generating function associated to the distributionof LISs, and key tothe analytic

attack on Ulam’s problem).

We also initiate the study of pattern avoidance in bipartite multigraphs and derive a generalized Gessel identity

for the number of bipartite 2-regular multigraphs avoiding a specific (monotone) pattern.

Keywords: Gessel’s identity, longest increasing sequence, random bipartite graphs, lattice walks.

1 Introduction

Let U and V henceforth denote two disjoint totally ordered sets (both ordered relations will be referred to by ?).

Typically, we will consider the case where |U| = |V| = n and denote the elements of U and V by u1,u2,...,unand

v1,v2,...,vnrespectively. Henceforth, we will always assume that the latter enumeration respects the ordered relation

inU orV, i.e., u1? u2? ... ? unand v1? v2? ... ? vn.

Let G = (U,V;E) denote a bipartite multi–graph with color classes U and V. Two distinct edges uv and u′v′of G

are said to be noncrossing if u and u′are in the same order as v and v′; in other words, if u ≺ u′and v ≺ v′or u′≺ u

and v′≺ v. A matching of G is called planar if every distinct pair of its edges is noncrossing. We let L(G) denote the

number of edges of a maximum size (largest) planar matching in G (note that L(G) depends on the graph G and on

the ordering of its color classes).

For the sake of simplicity we will concentrate solely in the case where |E| = rn and G is r–regular.

When r = 1, an r–regular multi–graph with color classes U and V uniquely determines a permutation. A planar

matching corresponds thus to an increasing sequence of the permutation, where an increasing sequence of length L

of a permutation π of {1,...,n} is a sequence 1 ≤ i1< i2< ... < iL≤ n such that π(i1) < π(i2) < ... < π(iL). The

∗Gratefully acknowledges the support of MIDEPLAN via ICM-P01–05, and CONICYT via FONDECYT 1010689 and FONDAP in Applied

Mathematics.

†Gratefully acknowledges the support of ICM-P01-05. This work was done while visiting the Depto. Ing. Matem´ atica, U. Chile.

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Longest Increasing Sequence (LIS) problem concerns the determination of the asymptotic, on n, behavior of the LIS

for a randomly and uniformly chosen permutation π. The LIS problem is also referred to as “Ulam’s problem” (e.g.,

in [Kin73, BDJ99, Oko00]). Ulam is often credited for raising it in [Ula61] where he mentions (without reference)

a “well–known theorem” asserting that given n2+1 integers in any order, it is always possible to find among them

a monotone subsequence of n+1 (the theorem is due to Erd˝ os and Szekeres [ES35]). Monte Carlo simulations are

reported in [BB67], where it is observed that over the range n ≤ 100, the limit of the LIS of n2+1 randomly chosen

elements, when normalized by n, approaches 2. Hammersley [Ham72] gave a rigorous proof of the existence of the

limit and conjectured it was equal to 2. Later, Logan and Shepp [LS77], based on a result by Schensted [Sch61],

proved that γ ≥ 2; finally, Vershik and Kerov [VK77] obtained that γ ≤ 2. In a major recent breakthroughdue to Baik,

Deift, Johansson [BDJ99] the asymptotic distribution of the LIS has been determined. For a detailed account of these

results, history and related work see the surveys of Aldous and Diaconis [AD99] and Stanley [Sta02].

1.1 Main Results

From the previous section’s discussion, it follows that one way of generalizing Ulam’s problem is to study the distri-

bution of the size of the largest planar matching in randomly chosen r–regular bipartite multi–graphs (for a different

generalization see [Ste77, BW88]). This line of research, originating in [KL02], turns out to be relevant for the

study of several other issues like the Longest Common Subsequence problem (see [KLM05]), interacting particle sys-

tems [Sep77], digital boiling [GTW01], and is directlyrelated to topics such as percolationtheory[Ale94] and random

matrix theory [Joh99].

In this article, we establish combinatorial identities which express gr(n;d) — the number of r-regular bipartite

multi–graphs with planar matchings with at most d edges — in terms of:

• The number of pairs of standard Young tableaux of the same shape and with a “descend-type” property (Theo-

rem 5).

• A signed sum of restricted lattice walks in Zd(Theorem 1).

Our arguments can be extended in order to characterize the distribution of the largest size of planar subgraphs of

randomly chosen r–regular bipartite multi–graphs (Theorem 4).

We also focus on the special case where d = 2 and r = 2 and try to gain insight into the behavior of gr(n;d)

i.e. the number of 2–regularbipartite multi-graphswhose largest planar matchings has at most 2 edges. Our method in

principle should work for general d and r. Although the special case where d is fixed a priori (independentof n) might

seem rather restricted, it is interesting by itself. Indeed, the determination of gr(n;d) for fixed d and r = 1 falls within

a very popular area of research referred to as pattern avoidance in permutations. Specifically, let σ be a permutationin

the symmetric group Snand say that “τ ∈ Smavoids σ as a subpattern” if there is no collection of indices 1 ≤ i1< i2<

... < in≤ m such that for 1 ≤ j,k ≤ n, τ(ij) < τ(ik) if and only if σ(j) < σ(k). Pattern avoidance concerns the study

of quantities such as the number of permutations avoiding a given patter σ, the asymptotic rate of such numbers, etc.

(see [B´04, Ch. 4] for a more in depth discussion of the pattern avoidance area). Note in particular that g1(n;d) equals

the number of permutations in Snavoiding the pattern (1,2,...,d +1). Well known results concerning avoidance of

patterns of length 3 thus imply that g1(n;2) is the n-th Catalan number [B´04, Corollary 4.7] and the g1(n;2)’s have

a simple generating function. Regev [Reg81] gives an asymptotic formula for g1(n;2). (Other results of enumerative

character on pattern avoidance in ordered graphs may be found in [BR01], in [EDSY07] and in [deM06].Other recent

results of enumerative character on restricted lattice paths may be found in [BF02], in [BM07] and in [Mis06].) We

obtain a formula (Theorem 8) for the generating function of the g2(n;2)’s. The identity we derive is a generalized

version of Gessel’s identity.

1.2 Models of Random Graphs: From k-regular Multi–graphs to Permutations

Most workon randomregulargraphsis based onthe so calledrandomconfigurationmodelof BenderandCanfield and

Bollob´ as [Bol85, Ch. II, § 4]. Below we follow this approach, but first we need to adapt the configurationmodel to the

bipartite graph scenario. GivenU,V, n and r as above, let U =U ×[r] andV =V ×[r]. An r–configuration ofU and

V is a one–to–onepairing ofU andV. These rn pairs are called edges of the configuration. Hence, a configurationcan

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be considered a graph, specifically, a perfect matching with color classesU andV. Moreover, viewing a configuration

as such bipartite graph enables us to speak also about its planar matchings (here the total ordering onU =U ×[r] and

V =V ×[r] is the lexicographic one induced by ? and ≤).

The natural projection ofU =U ×[r] andV =V ×[r] ontoU andV respectively (ignoring the second coordinate)

projects each configuration F to a bipartite multi–graph π(F) with color classes U andV. Note in particular that π(F)

may contain multiple edges (arising from sets of two or more edges in F whose end–points correspond to the same

pair of vertex in U and V). However, the projection of the uniform distribution over configurations of U and V is

not the uniform distribution over all r–regular bipartite multi–graphs onU andV (the probability of obtaining a given

multi–graphis proportionalto a weight consisting of the productof a factor 1/j! for each multiple edge of multiplicity

j). Since a configuration F can be considered a graph, it makes perfect sense to speak of the size L(F) of its largest

planar matching.

We denote an element (u,i) ∈U by uiand adopt an analogous convention for the elements of V. We shall further

abuse notation and denote by ? the total order on U given by ui? ˜ ujif u ≺ ˜ u or u = ˜ u and i ≤ j. We adopt a similar

convention forV.

Let Gr(U,V;d) denote the set of all r–regular bipartite multi–graphs onU andV whose largest planar matching is

of size at most d. Note that if |U| = |V| = n, then the cardinality of Gr(U,V;d) depends onU andV solely through n.

Thus, for |U| = |V| = n, let gr(n;d) = |Gr(U,V;d)|.

The first step in our considerations is an identification of Gr(U,V;d) with a subset of configurations of U and V.

Specifically, we associate to an r–regular multi–graph G = (U,V;E) the r–configuration G of U and V such that

π(G) = G where: If (u,v) is an edge of multiplicity t in G for which there are i edges (u,v′) in G such that v ≺ v′,

and j edges (u′,v) in G such that u ≺ u′, then for every s ∈ [t], the pairing (ui+s,vj+t−s+1) belongs to G. Note that the

number of edges of G equals the number of edges of G.

Let Gr(U,V;d) be the collection of configurationsG associated to some G ∈ Gr(U,V;d). Observe, that gr(n;d) =

|Gr(U,V;d)|.

For an edge (u,v) we say that M ⊆?u′∈U : u′? u?×?v′∈V : v′? v?is a planar matching that ends with (u,v)

we speak of a largest planar matching of G up to u (or v) in order to refer to a largest planar matching that ends with

edge (u,v).

Note that the way in which G is derived from G, implies in particular that for u ∈ U and i ≤ j, the size of the

maximum planar matching in G using nodes up to uiis at least as large as the size of the maximum planar matching

using nodes up to uj. A similar fact holds for elements v ∈V.

Several of the concepts introduced in this section are illustrated in Figure 1.

if the edges in M are non–crossing and (u,v) ∈ M. Since there is a unique edge incident to every node in G, say (u,v),

1.3Young tableaux

A (standard) Young tableau of shape λ = (λ1,...,λr) where λ1≥ λ2≥ ... ≥ λr≥ 0, is an arrangement T = (Tk,l) of

λ1+...+λrdistinct integers in an array of left–justified rows, with λielements in row i, such that the entries in each

row are in increasing order from left to right, and the entries of each column are increasing from top to bottom (here

we follow the usual convention that considers row i to be above row i+1). One says that T has r rows and c columns

if λr> 0 and c = λ1respectively. The shape of T will be henceforth denoted shp(T) and the collection of Young

tableau with entries in the set S and with at most d columns will be denoted T(S;d).

The Robinson correspondence (rediscovered independently by Schensted) states that the set of permutations of

[m] is in one to one correspondence with the collection of pairs of equal shape tableaux with entries in [m]. The

correspondence can be constructed through the Robinson–Schensted–Knuth (RSK) algorithm — also referred to as

row–insertion or row–bumping algorithm. The algorithm takes a tableau T and a positive integer x, and constructs a

new tableau, denoted T ← x. This tableau will have one more box than T, and its entries will be those of T together

with one more entry labeled x, but there is some moving around, the details of which are not of direct concern to us,

except for the following fact:

Lemma 1 [Bumping Lemma [Ful97, pag. 9]] Consider two successive row–insertions, first row inserting x in a

tableau T and then row–inserting x′in the resulting tableau T ← x, given rise to two new boxes B and B′as shown in

Figure 2.

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v1

v2

(a)

v3

u1

u3

u2

v1

1

v2

1

u1

1

u1

2

u2

1

u2

2

u1

3

v2

2

v1

3

v2

3

u2

3

v1

2

(b)

Figure 1: (a) A 2–regular multi–graph G. (b) Configuration G associated to G.

• If x ≤ x′, then B is strictly left of and weakly below B′.

• If x > x′, then B′is weakly left of and strictly below B.

Given a permutation π of [m], the Robinson–Schensted–Knuth (RSK) correspondence constructs (P(π),Q(π)) such

that shp(P(π)) = shp(Q(π)) by,

• starting with a pair of empty tableaux, repeatedly row–inserting the elements π(1),...,π(n) to create P(π), and,

• placingthe value i into the box of Q(π)’s diagram correspondingto the box created duringthe i–th insertion into

P(π).

Two remarkable facts about the RSK algorithm which we will exploit are:

Remark 1 [RSK Correspondence[Ful97, pag.40]]TheRSK correspondencesets up aone–to–onemappingbetween

permutations of [m] and pairs of tableaux (P,Q) with the same shape.

Remark 2 [Symmetry Theorem [Ful97, pag. 40]] If π is a permutation of [m], then P(π−1) = Q(π) and Q(π−1) =

P(π).

Moreover, it is easy to see that the following holds:

Remark 3 Let π be a permutation of [m]. Then, π has no ascending sequence of length greater than d if and only if

P(π) and Q(π) have at most d columns.

The reader interested on an in depth discussion of Young tableaux is referred to [Ful97].

1.4 Walks

We say that w = w0...wmis a lattice walk in Zdof length m if ||wi−wi−1||1= 1 for all 1 ≤ i ≤ m. Moreover, we say

that w starts at the origin and ends in ? p if w0=?0 and wm=? p. For the rest of this paper, all walks are to be understood

as lattice walks in Zd. Let W(d,m;? p) denote the set of all walks of length m from the origin to ? p ∈ Zd.

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B

x ≤ x′

B′here

x > x′

B′here

Figure 2: New tableau entries created through row–insertions.

We will often identify the walk w =w0···wmwith the sequence d1...dmsuch that wi−wi−1=sign(di)? e|di|, where

? ejdenotes the j–th element of the canonical basis of Zd. If diis negative, then we say that the i–th step is a negative

step in direction |di|, or negative step for short. We adopt a similar convention when diis positive.

We say that two walks are equivalent if both subsequences of the positive and the negative steps are the same. For

each equivalence class consider the representative for which the positive steps precede the negative steps. Each such

representative walk may hence be written as a1a2···|b1b2··· where the ai’s and bj’s are all positive. For an arbitrary

collection of walks W, all with the same number of positive and the same number of negative steps, we henceforth

denote byW∗the collection of the representative walks inW.

Recall that one can associate to a permutation π of [d] the Toeplitz point T(π) = (1−π(1),...,d −π(d)). Note

that in a walk from the origin to a Toeplitz point, the number of steps in a positive direction equals the number of steps

in a negative direction. In particular, each such walk has an even length.

In cases where we introduce notation for referring to a family of walks from the origin to a given lattice point ? p,

such as W(d,m;? p), we sometimes consider instead of ? p a subset of lattice points P. It is to be understood that we are

thus making reference to the set of all walks in the family that end at a point in P. A set of lattice points of particular

interest to the ensuing discussion is the set of Toeplitz points, henceforth denoted T.

We now come to a simple but crucial observation: there is a natural identification ofU ×[r] with [rn] that respects

the total order in each of these sets (? in the former and ≤ in the latter). A similar observation holds for V ×[r].

Hence, when m = rn the sequences of positive and negative steps in a walk inW(d,2m;T) can be referred to as:

au1

1···aur

1au1

2···aur

2···au1n···aurn

and

bv1

1···bvr

1bv1

2···bvr

2···bv1n···bvrn.

Let W′(d,2m;T(π)) be the set of all walks in W∗(d,2m;T(π)) whose positive steps au1

bv1

1···aurnand negative steps

1···bvrnsatisfy: aui ≥ aui+1 and bvi ≥ bvi+1 for all u ∈U, v ∈V and 1 ≤ i < r.

2Counting Planar Mathchings and Planar Subgraphs

We are now ready to state the main result of this paper.

Theorem 1 For every positive integers n,d,r,

gr(n;d) =∑

π

sign(π)??W′(d,2rn;T(π))??.

Our proofof Theorem1 is stronglybased onthe argumentsused in [GWW98] to provethe followingresult concerning

1–regular bipartite graphs:

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Theorem 2 The signed sum of the number of walks of length 2m from the origin to Toeplitz points is?2m

This last theorem gives a combinatorial proof of the following well known result:

m

?times the

number um(d) of permutations of length m that have no increasing sequence of length bigger than d.

Theorem 3 [Gessel’s Identity] Let Iν(t) denote the Bessel function of imaginary argument, i.e.

Iν(t) =∑

m≥0

1

m!Γ(m+ν+1)

?x

2

?2m+ν

.

Then,

∑

m≥0

um(d)

m!2x2m= det(I|r−s|(2x))r,s=1,...,d.

We now describe a random process which researchers have studied, either explicitly or implicitly, in several different

contexts. Let Xi,jbe a non–negativerandom variable associated to the lattice point (i, j) ∈ [n]2. ForC ⊆ [n]2, we referr

to ∑(i,j)∈CXi,jas the weight of C. We are interested on the determination of the distribution of the maximum weight

ofC over allC = {(i1, j1),(i2, j2),...} such that i1,i2,... and j1, j2,... are strictly increasing.

Johansson [Joh99] considered the case where the Xi,js are independent identically distributed according to a geo-

metric distribution. Sep¨ al¨ ainen [Sep77] and Gravner, Tracy and Widom [GTW01] studied the case where the Xi,js are

independent identically distributed Bernoulli random variables (but, in the latter paper, the collections of lattice points

C = {(i1, j1),(i2, j2),...} were such that i1,i2,... and j1, j2,... were weakly and strictly increasing respectively).

The main result of this paper, i.e., Theorem 1, says that if (Xi,j)(i,j)∈[n]2 is uniformly distributed over all adjacency

matrices of r–regular multi–graphs, then the distribution of the maximum weight evaluated at d can be expressed as

a signed sum of restricted lattice walks in Zd. A natural question is whether a similar result holds if one relaxes the

requirement that the sequences i1,i2,... and j1, j2,... are strictly increasing. For example, if one allows them to be

weakly increasing. This is equivalent to asking for the distribution of the size of a planar subgraph, i.e., the largest set

of non–crossing edges which may share endpoints in a uniformly chosen r–regular multigraph. A line of argument

similar to the one we will use in the derivation of Theorem 1 yields:

Theorem 4 Let ˆ g(n;d) be the number of r-regular bipartite multi–graphs with no larger than d set of non–crossing

edgeswhichmay shareendpoints. Then, ˆ g(n;d) equalsthenumberofpairs of equalshapeYoungtableauxin T([rn];d)

satysifying:

Condition (ˆT): If for each i ∈ [n] and 1 ≤ s < r, the row containing r(i−1)+s+1 is weakly above the

row containing r(i−1)+s.

Moreover,

ˆ g(n;d) =∑

π

sign(π)

???? W′(d,2rn;T(π))

??? ,

where? W′(d,2rn;T(π)) is the set of all walks in W∗(d,2rn;T(π)) whose positive steps au1

In the rest of the paper we give two independent proofs of Theorem 1.

1···aurnand negatives steps

bv1

1···bvrnsatisfy: aui < aui+1 and bvi < bvi+1 for all u ∈U, v ∈V and 1 ≤ i < r.

3 First Proof

Let m = rn. Recall that Gr(U,V;d) can be thought of as a collection of permutations of [m]. Thus, we may think of

the RSK correspondence as being defined over Gr(U,V;d). In particular, for an r–configuration F of U and V we

may write (P(F),Q(F)) to denote the pair of Young tableaux associated to the permutationdetermined by F. Figure 3

shows the result of applying the RSK algorithm to an r–configuration.

We say that a Young tableau in T([m];d) satisfies

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1

2

4

6

3

5

1

2

3

4

5

6

Figure 3: Pair of Young tableaux associated through the RSK algorithm to the 2–configuration of Figure 1.b.

Condition (T): If for each i ∈ [n] and 1 ≤ s < r, the row containing r(i−1)+s is strictly above the row

containing r(i−1)+s+1.

The following result characterizes the image of Gr(U,V;d) through the RSK correspondence.

Theorem 5 The number gr(n;d) equals the number of pairs of equal shape tableaux in T([m];d) satisfying condition

(T). Specifically, the RSK correspondence establishes a one–to–one correspondance between Gr(U,V;d) and the

collection of pairs of equal shape tableaux in T([m];d) satisfying condition (T).

Proof:

T([m];d). Corollary 1 implies that, for every i ∈ [n] the row insertion process through which P(G) is built is such

that the insertion of the values r(i−1)+1,...,ri gives rise to a sequence of boxes each of which is strictly below the

previous one. This implies that Q(G) satisfies condition (T).

We still need to show that P(G) also satisfies condition (T). For G ∈ Gr(U,V;d) let the transpose of G, denoted

GT, be the bipartite graph over color classes U and V such that uivj is an edge of GTif and only if ujvi is an

edge of G. Note that GT∈ Gr(U,V;d) if and only if G ∈ Gr(U,V;d). A direct consequence of Remark 2 is that

(P(GT),Q(GT)) = (Q(G),P(G)). Hence, P(G) must also satisfy condition (T).

Suppose now that (P,Q) is a pair of equal shape tableaux in T([m];d) both of which satisfy condition (T). Let F be an

r–configuration of U and V such that (P(F),Q(F)) = (P,Q) (here we identify us

of [m]). The existence of F is guaranteed by Remark 1. Remark 3 implies that F’s largest planar matching is of size

at most d. Moreover, since Q(F) satisfies property (T), Lemma 1 implies that the edges of F incident to us

cross. Similarly, one can conclude that the edges fo F incident to vs

Gr(U,V;d).

Let G ∈ Gr(U,V;d)). Remark 3 implies that P(G) and Q(G) are tableaux of equal shape that belong to

iand vs

iand view F as a permutation

iand us+1

i

iand vs+1

i

cross. It follows that F belongs to

Example 1 Note that condition (T), as guaranteed by Theorem 5, is reflected in the tableaux shown in Figure 3 (for

the tableau in the left; 4, 1 and 3 are strictly above 6, 2 and 5 respectively, while for the tableau in the right; 1, 3 and

5 are strictly above 2, 4 and 6 respectively).

For a walk w = a1···am|b1···bmin W′(d,2m;T(π)) let ˜ w = ˜ a1··· ˜ am|˜b1···˜bmbe such that ˜ ai= aiand˜bi= bm−i.

Denote by? W(d,2m;T(π)) the collection of all ˜ w for which w belongs to W′(d,2m;T(π)). Our immediate goal is to

Theorem 6 There is a bijection between Gr(U,V;d) and the walks in? W(d,2m;?0) staying in the region x1≥ x2≥

We now discuss how to associate walks to Young tableaux. First we need to introduce additional terminology. We say

that a walk w = a1···amsatisfies

establish the following

... ≥ xd.

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?0

Figure 4: Walk in? W(d,2m;?0) associated to the pair of Young tableaux of Figure 3 (and thus also to the graph of

Figure 1).

Condition (W): If for each i ∈ [n] and 1 ≤ s < r it holds that ar(i−1)+s≥ ar(i−1)+s+1.

Let ϕ be the mapping from T([m];d) to walks in W(d,m;Zd) such that ϕ(T) = a1···amwhere aiequals the column

in which entry i appears in T. It immediately follows that:

Lemma 2 The mapping ϕ is a bijection between tableaux in T([m];d) satisfying condition (T) and walks of length

m starting at the origin, moving only in positive directions, staying in the region x1≥ x2≥ ··· ≥ xdand satisfying

condition (W).

Proof: If ϕ(T) = ϕ(T′) for T,T′∈ T([m];d), then T and T′have the same elements in each of their columns. Since

in a Young tableau the entries of each column are increasing from top to bottom, it follows that T = T′. We have thus

established that ϕ is an injection.

Assume now that w = a1···amis a walk of length m starting at the origin, moving only in positive directions, staying

in the regionx1≥x2≥···≥xdandsatisfyingcondition(W). Denote byC(l) the set of indices j for whichaj=l. Note

that since w is a walk in Zd, then C(l) is empty for all l > d. Let T be the Young tableau whose l–th column entries

correspond to C(l) (obviously ordered increasingly from top to bottom). Note that T is indeed a Young tableau since

|C(1)| ≥ |C(2)| ≥ ... ≥ |C(d)| and given that the entries on each row of T are strictly increasing (the latter follows

from the fact that w stays in the region x1≥ x2≥ ··· ≥ xd). Observe that T belongs to T([m];d). We claim that T

satisfies condition (T). Indeed, by construction and since w satisfies condition (W), for each i ∈ [n] it must hold that

the indices of the columns of the entries r(i−1)+1,...,ri of T is a weakly decreasing sequence. Hence, for every

1 ≤ s < r, the entry r(i−1)+s+1 is weakly to the left of r(i−1)+s. Since r(i−1)+s+1> r(i−1)+s and T is a

tableau, it must be the case that the entry r(i−1)+s+1 is strictly below the entry r(i−1)+s.

Note that if T and T′belong to T([m];d) and have the same shape, then ϕ(T) and ϕ(T′) are walks that terminate at

the same lattice point.

Corollary 1 There is a bijectionbetweenorderedpairs of tableauxofthe same shapebelongingtoT([m];d) satisfying

condition (T), and walks in? W(d,2m;?0) staying in the region x1≥ x2≥ ... ≥ xd.

Proof: By Lemma 2 there is a bijection between ordered pairs of tableaux with the claimed properties and ordered

pairs of walks of length m starting at the origin, moving only in positive directions that stay in the region x1≥ x2≥

...≥xdandsatisfy condition(W).Saysuchpairofwalksarec1···cmandc′

is the sought after walk with the desired properties.

1···c′mrespectively. Then,c1···cm|c′m···c′

1

Figure4 illustrates the bijectionimplicit in the proofofCorollary1. Note thatTheorem6is an immediateconsequence

of Theorem 5 and Corollary 1.

Proof: [of Theorem 1] The desired conclusion is an immediate consequence of Theorem 6 and the existence of a a

parity-reversinginvolutionρ on the walks w in? W(d,2m;?0) not staying in the regionx1≥x2≥...≥xd. The involution

is most easily described if we translate the walks to start at (d−1,d−2,...,0); the walks are then restricted not to lie

Page 9

?0

Figure 5: Walk Φ(G) = 111122|112121 for the multi–graph G of Figure 1 (direction 1 is to the right and direction 2

is up — negative steps are represented by segmented lines.)

completely in the region R defined by x1> x2> ... > xd. Let N be the subset of the translated walks of? W(d,2m;?0)

segment of c1...ctof w terminates in a vertex (p1,...,pd) ?∈ R. Hence, there is exactly one j such that pj= pj+1.

Walk ρ(w) is constructed as follows:

not lying completely in R. Let w = c1...c2m∈ N and let t be the smallest index such that the walk given by the initial

• Leave segment c1...ctunchanged.

• For each i ∈ [2n], define S(i) = {s ∈ [2m] : r(i−1) < s ≤ ri}, S0(i) = {s ∈ S(i) : s >t,cs= j} and S1(i) =

{s ∈ S(i) : s >t,cs= j+1}. For i ≤ n (respectively i > n), assign the value j+1 to the |S0(i)| first (respec-

tively last) coordinates of (cs: s ∈ S0(i)∪S1(i)) and the value j to the remaining |S1(i)| coordinates.

It is easy to see that if w terminates in (q1,...,qd), then ρ(w) terminates in (q1,...,qj+1,qj,...,qd). Hence, ρ reverses

the parity of w. Moreover,ρ◦ρ is the identity. It remains to show that ρ(w) ∈N. Obviouslyρ(w) does not stay in R (as

w does not). Hence,it sufficesto showthe following: ifρ(w)=a1...am|b1...bm, thenforeachi∈[n] and1≤s<r we

have ar(i−1)+s≥ ar(i−1)+s+1and br(i−1)+s≤ br(i−1)+s+1. This is clearly true for every block {r(i−1)+s : 1 ≤ s ≤ r}

completelycontainedinsidew’s unchangedsegment(i.e.,1,...,t)andinside w’s modifiedsegment(i.e.,t +1,...,2m),

giventhat it is true forw and by the definitionof ρ. Thereis still the case to handlewheret ∈{r(i−1)+s : 1 ≤ s ≤ r}.

Here, it is true by the following observation: if t ≤ m then ct= j+1, otherwise ct= j.

4 Second proof

Henceforth let m = rn. In this section we introduce two mappings Φ and φ. The former is shown to be an injection

that, when restricted to F = Gr(U,V;d), takes values in W′(d,2m;?0). Our first goal is to characterize those walks

that belong to Φ(F ). The second mapping φ plays a crucial role in fulfilling this latter objective. Then, relying on

the aforementioned characterization we define a parity reversing involution on W′(d,2m;T)\Φ(F ). This essentially

establishes Theorem 1.

Let Φ be the function that associates to an r–configuration F of U and V the value Φ(F) = au1

W∗(d,2m;Zd), where

1···aurn|bv1

1···bvrn∈

• auequals the largest size of a planar matching of F using nodes up to u,

• bvequals the largest size of a planar matching of F using nodes up to v.

Note that indeed Φ(F) ∈ W∗(d,2m;Zd) when F is an r–configuration of U and V. Figure 5 illustrates the definition

of Φ(·).

The following definition will be instrumental in the introduction of a mapping between walks and configurations.

Page 10

B

A

Figure 6: Crossing obtained from left to right ordered sets A and B.

Figure 7: The quasi configuration φ(w) associated to w = 112122|122122. Continuous lines corresponds to the cross-

ing of A1(w) and B1(w) and segmented lines to the crossing of A2(w) and B2(w).

Definition 1 Let A and B be two linearly ordered sets of equal size. We say that a quasi configurationis obtainedfrom

A and B in a crossing way if the first element of A is paired with the last element of B, and so on, until finally the last

element of A is paired to the first element of B.

Figure 6 illustrates the concept just introduced. We say that H is a quasi r–configuration of U and V if it can be

obtained from a configuration F of U and V by “breaking” (deleting) some of its “edges” (pairings). Note that the

same quasi r–configuration may be obtained by “breaking” different r–configurations.

For w = au1

ready to introduce a mapping between walks and quasi configurations. Let φ be a function that associates to a walk

w ∈ W∗(d,2m;Zd) a quasi r–configuration φ(w) as follows: for each k, if |Ak(w)| ≥ |Bk(w)| then connect the initial

segment of Ak(w) of size |Bk(w)| in a crossing way with Bk(w). If |Ak(w)| ≤ |Bk(w)|, then connect the terminal

segment of Bk(w) of size |Ak(w)| in a crossing way with Ak(w). Figure 7 illustrates φ(·)’s definition.

Fact 1 Let w ∈ W∗(d,2m;Zd). Then, the edges in φ(w) incident to two distinct elements of Ak(w) must cross. A

similar observation holds for Bk(w).

1···aurn|bv1

1···bvrn∈ W∗(d,2m;Zd), let Ak(w) = {u : au= k} and Bk(w) = {v : bv= k}. We are now

Fact 2 Let w = Φ(F) for an r–configuration F of U and V and let (u,v) be a pairing of F. Then, u ∈ Ak(w) if and

only if v ∈ Bk(w).

The following result gives an interpretation in terms of graphs of what it means for a walk starting at the origin to

terminate also at the origin.

Lemma 3 Let w ∈ W∗(d,2m;Zd). Then, φ(w) is an r–configuration of U and V if and only if w ∈ W∗(d,2m;?0) for

some d.

Proof: A closed walk w passes through the origin if and only if |Ak(w)| = |Bk(w)| for all k. The latter is certainly

equivalent to φ(w) being a configuration.

We now prove a technical result.

Page 11

Lemma 4 Let k ∈ [d] be arbitrary. For every r–configuration F of U and V, the set of edges incident to Ak(Φ(F))

equals the set of edges incident to Bk(Φ(F)).

Proof:

v ∈ Bk(Φ(F)).

The following result establishes that Φ(·) is an injection.

Lemma 5 For every r–configurationF ofU andV, it holds that φ(Φ(F)) = F.

Let u ∈ Ak(Φ(F)). There is a unique v such that (u,v) is a pairing of F. By Fact 2, it must hold that

Proof: By Fact 1 and Lemma 4, Ak(Φ(F)) and Bk(Φ(F)) are equal size sets that must be joined in the crossing way

in F. Since a pairing of F is an element of Ak(Φ(F))×Bk(Φ(F)) for some k, it follows that φ(Φ(F)) = F.

Lemma 6 Let F be a family of r–configurations of U andV. A walk w belongs to Φ(F ) if and only if Φ(φ(w)) = w

and φ(w) ∈ F .

Proof: If φ(w) ∈ F , then w = Φ(φ(w)) belongs to Φ(F ). If w = Φ(F) for some r–configuration F ofU andV, then

Lemma 5 implies that φ(w) = F. If in addition F ∈ F , then one gets that φ(w) ∈ F .

Two walks inW∗(d,2m;Zd) are certainly equal if their sequence of positive and negativesteps agree. The next lemma

gives a simpler necessary and sufficient condition for the equality of two walks w and Φ(φ(w)) when w is a closed

walk that goes through the origin. Indeed, it says that one only needs to focus on establishing the equality of the

sequence of their positive steps. The result will be useful later in order to establish the equality of two walks w and

Φ(φ(w)).

Lemma 7 Let w ∈W∗(d,2m;?0). Then, Φ(φ(w)) and w agree in their positive steps if and only if Φ(φ(w)) = w.

Proof: If Φ(φ(w)) = w, then Φ(φ(w)) and w clearly agree in their positive steps. To prove the converse, let w′=

Φ(φ(w)). Assume w′and w agree in their positive steps. First, recall that by Lemma 3, φ(w) is an r–configuration of

U and V. Hence, Lemma 5 implies that φ(w′) = φ(Φ(φ(w))) = φ(w). Thus, Ak(w) = Ak(w′) for every k. Since φ(w′)

and φ(w) are the same configurations, they have the same set of edges. Consider v ∈ Bk(w). There is a unique edge

(u,v) of φ(w) incident on v. By Fact 2, we have that u ∈ Ak(w) = Ak(w′). But edge (u,v) is an edge of φ(w′). Hence,

again by Fact 2, we get that v ∈ Bk(w′). We have shown that Bk(w) ⊆ Bk(w′). The reverse inclusion can be similarly

proved. Since k was arbitrary, we conclude that the negative steps of w and w′are the same, and the two walks must

thus be equal.

For the walk w = au1

?u′∈U : u′? u?respectively.

Example 2 For the walk 111122|112121of Figure 5 we have:

1···aurn|bv1

1···bvrn, denote by k(u) and l(u) the number of occurrences of auand au−1 in

u

1

1

0

2

2

0

3

3

0

4

4

0

5

1

4

6

2

4

k(u)

l(u)

We say that w satisfies

Condition (C): If for each u such that au> 1, l(u) > 0 and the l(u)–th-to-last appearance of au−1 in

the negativesteps of w, if it exists, comes before the k(u)–th-to-last appearance of auin the negativesteps

of w.

Lemma 8 Let w=au1

ofU andV if and only if w satisfies condition (C).

1···aurn|bur

1···bvrnbea walk inW∗(d,2m;T). Then, Φ(φ(w)) =w andφ(w) is anr–configuration

Page 12

Proof:

the other hand, if w satisfies condition (C), then w also needs to terminate at the origin. Hence, φ(w) would be an

r–configuration ofU andV. Indeed, let j be the smallest coordinate in which the terminal point of w is positive. Note

that j >1 by the definition of Toeplitz points. Let u be maximumso that au= j. By the choice of j, there will be fewer

than k(u) appearances of j and at least l(u) appearances of j−1 among the negative steps. This contradicts the the

fact that w satisfies condition (C). We thus can assume without loss of generality that w starts and ends at the origin.

Let Φ(φ(w)) =a′

u1

vr

step. Assume that au= a′ufor each u ≺ ˜ u and a˜ u?= a′˜ u. We claim that a′˜ u≤ a˜ u. Indeed, suppose this is not the case.

Since a′˜ uequals the size of the largest planar matching of φ(w) up to ˜ u, there is a u ≺ ˜ u such that a′u= a˜ uand the edges

incident to u and ˜ u of φ(w) are non-crossing. We also have a′u=auby the choice of ˜ u. Hence, u and ˜ u belong to Ak(w)

for some k. By Fact 1 the edges incident to u and ˜ u must be non-crossing. A contradiction. This establishes our claim.

It follows that a′˜ u= a˜ uif and only if a′˜ u≥ a˜ u. We now establish a condition equivalent to a′˜ u≥ a˜ uby considering the

following two cases:

Since φ(w) is an r–configuration of U and V, Lemma 3 implies that w must terminate at the origin. On

1···a′

urn|b′

1···b′

vrn. By Lemma7, Φ(φ(w)) andw aredistinct ifandonlyif theydifferin somepositive

• Case a˜ u= 1: Then, certainly a′˜ u≥ a˜ u.

• Case a˜ u> 1: Then, there is a u ≺ ˜ u such that a′u= a˜ u−1 and the edges incident to u and ˜ u are non–crossing in

φ(w). So we can extend with the edge incident to ˜ u the size a′uplanar matching of φ(w) up to u. Thus, it must

be the case that a′˜ u≥ a˜ u.

Summarizing a˜ u= a′˜ uif and only if

• a˜ u= 1, or

• if a˜ u> 1 and there is a u ≺ ˜ u such that a′u= a˜ u−1 and the edges ˜ u and u are non–crossing in φ(w).

The lemma follows by observing that when a˜ u> 1, the fact that w satisfies condition (C) amounts to saying that there

is a u ≺ ˜ u such that au= a˜ u−1 and the edges incident to u and ˜ u are non–crossing in φ(w). So, all positive steps of

Φ(φ(w)) and w agree if and only if for each u such that au> 1, l(u) > 0 and the l(u)–th-to-last appearance of au−1

in the negative steps of w, if it exists, comes before the k(u)–th-to-last appearance of auin the negative steps of w.

So far in this section we have not directly being concerned with walks W′(d,2m;T) nor the collection of configu-

rations Gr(U,V;d). The next result is the link through which we use all previous results in order to prove Theorem 1.

Lemma 9 Let w ∈W′(d,2m;?0). If Φ(φ(w)) = w, then φ(w) ∈ Gr(U,V;d).

Proof: Suppose Φ(φ(w)) = w and w = au1

that since w is a closed walk that goes through the origin, by Lemma 3, we have that φ(w) is an r–configuration of U

andV. Thus, it must be the case that either there is a u ∈U such that for some s <t the edges incident to usand utare

non–crossing, or there is a v ∈ V such that for some s < t the edges incident to vsand vtare non–crossing. Without

loss of generality assume the former case holds. It follows that, the largest planar matching up to usis strictly smaller

than the largest planar matching up to ut, i.e., aus < aut. This contradicts the fact that w belongs to W′(d,2m;?0).

1···aurn|bv1

1···bvrnis such that φ(w) does not belong to Gr(U,V;d). Note

Theorem 7 The mapping Φ is a bijection between Gr(U,V;d) and the collection of walks in W′(d,2m;T) satisfying

condition (C).

Proof: By Lemma 5 we know that Φ is an injection. We claim it is also onto. Indeed, if w is a walk in W′(d,2m;T)

satisfying condition (C), then Lemmas 3, 6, 8, and 9 imply that φ(w) belongs to Gr(U,V;d) and Φ(φ(w)) = w.

Proof: [of Theorem 1] The desired conclusion is an immediate consequence of Theorem 7 and the existence of a

parity-reversing involution ρ on walks w in W′(d,2m;T) that don’t satisfy condition (C). To define ρ, assume w =

Page 13

au1

the l(u)–th-to-last occurrence of au−1 among the negative steps; if l(u) = 0 then let v = rn+1.

Walk ρ(w) is constructed as follows:

1···aurn|bv1

1···bvrnand let u be the smallest index for which w does not satisfy condition (C). Let v be such that bvis

• Leave segments au1

1···auand bv···bvrnunchanged.

?

• For every i ∈ [n], let S0(i) =

to the |S1(i)| first coordinates in (aus

The application of ρ does not change the smallest index not satisfying the sufficient condition of Lemma 8. It follows

that ρ(w) also violates condition (C). We claim that ρ(w) = a′

to show that for each i ∈ [n] and 1 ≤ s < r, we have a′

{us

two cases to consider are u = us

the details to the reader.

Assume w terminates at T(π) for some permutation π of [d]. Let τ be a transposition of auand au−1. Finally, we

claim that ρ(w) terminates in T(π◦τ). Indeed, by our choice of u, the number of appearances of auin bv···bvrnis less

than k(u). It must equal to k(u)−1, otherwise we could have chosen the index of the (k(u)−1)–th appearance of au

for u. Hence, in the unchanged segments of the walk w, there is one net positive step in direction auand zero net steps

in direction au−1. It follows that in the segment of w that changes, there are au−π(au)−1 and au−1−π(au−1)

net positive steps in directions auand au−1 respectively. Let σsdenote the s–th coordinate of the terminal point of

a walk σ. We get that ρ(w)au= au−1−π(au−1)+1 = au−π(au−1) and similarly ρ(w)au−1= au−π(au)−1 =

au−1−π(au).

s : aus

i= au,u ≺ us

i:s ∈ S0(i)∪S1(i)) and the value au−1 to the remaining|S0(i)| coordinates.

i

?

and S1(i) =

?

s : aus

i= au−1,u ≺ us

i

?

. Assign the value au

u1

1···a′

and b′

urn|b′

vs

v1

1···b′

i≥ b′

vrnbelongs to W′(d,2m;T). We need

. This is clearly true for every block

us

i≥ a′

us+1

i

vs+1

i

i: s ∈ [r]} completely contained in the unchanged segments, and also inside the modified segment. The remaining

iand/or v = vs′

jfor some s < r and/or s′> 1. Both cases are easy to handle. We leave

5 The d = r = 2 Case

Our objective throughoutthis section is to initiate the study of pattern avoidance in bipartite multi–graphs. We start by

addressing the case of 2-regular bipartite multi-graphs where the pattern to be avoided is the one shown in Figure 8.

Figure 8: Multi-graph subpattern to avoid.

We recall that W∗(2,m;? p) is the set of all walks to ? p of length m whose positive steps au1

negative steps bv1

which in addition satisfy: au1 ≥ au2 and bv1 ≥ bv2 for all u ∈U, v ∈V.

In this section it will be convenient to denote by W∗

steps and m−negative steps, and m = m++m−. We will abide by a similar convention when referring toW′(d,m;? p).

We also denote byW′′

ative steps, the resulting walk belongs toW′

and by w′′

?m++m−

1···au2nprecede the

1···bv2swhere m = 2n+2s. Furthermore, recall that W′(2,m;? p) is the collection of all such walks

d(m+,m−;? p) the set of walks of W∗(d,m;? p) with m+positive

d(·) the set of walks w such that after rearrangingw so that the positive steps precede the neg-

d(·). Finally, we denote by w′

d(m+,m−;? p) the cardinality ofW′′

?

d(m+,m−;? p) the cardinalityofW′

d(m+,m−;? p)

d(m+,m−;? p). Hence, for instance

w′

d(m+,m−;? p)

m+

= w′′

d(m+,m−;? p).

Page 14

Henceforth assume m+and m−are even. Let W′

W′

2(K,L;m+,m−;? p), K ⊆U and L ⊆V, denote the set of the walks of

2(m+,m−;? p) such that

• if u ∈ K, then au1 < au2 (or equivalently, au1 = 1 and au2 = 2), and

• if v ∈ L, then bv1 < bv2 (or equivalently, bv1 = 1 and bv2 = 2).

Let w′

otherwise. We start by establishing a relation between the terms w′

quantities w′

by (a)bwhere a and b are non–negativeintegers.

Lemma 10 For very ? p ∈ Z2, and m+,m−∈ 2N,

w′

m+!m−!

k,l≥0

Moreover, the same identity holds when w′

2(k,l;m+,m−;? p) take the value ∑K⊆U:|K|=k ∑L⊆V:|L|=l|W′

2(K,L;m+,m−;? p)|, if k ≤ m+/2 and l ≤ m−/2, and 0

2(m+,m−;? p) and the somewhat more manageable

2(k,l;m+,m−;? p). The relation makes use of the standard convention of denoting a(a−1)···(a−b+1)

2(m+,m−;? p)

=∑

(−1)k+lw′

2(k,l;m+,m−;? p)

m+!m−!

=∑

k,l≥0

2.

(−1)k+lw′

2(k,l;m+,m−;? p)

(m+−2k)!(m−−2l)!×

1

(m+)2k(m−)2l

.

2is replaced by w′′

Proof: We will prove only the first equality. The other identities are elementary. First observe that

W′

2(m+,m−;? p)=

W∗(m+,m−;? p)\

?

[

u∈U

W′

2({u}, / 0;m+,m−;? p)∪

[

v∈V

W′

2(/ 0,{v};m+,m−;? p)

?

Hence, by the Principle of Inclusion-Exclusion,

w′

2(m+,m−;? p)=

∑

K⊆U∑

k≥0∑

L⊆V

(−1)|K|+|L|??W′

(−1)k+lw′

2(K,L;m+,m−;? p)??

= ∑

l≥0

2(k,l;m+,m−;? p).

The desired conclusion follows immediately.

Clearly, there are walks in Z from the origin to p ∈ Z with m+positive steps preceding m−negative steps if and only

if m+−m−1= p. Moreover, there is at most one such walk. Since w′

we have

?

0,

Also note that

w′′

m+

1(m+,m−;p) denotes the number of these walks,

w′

1(m+,m−;p) =

1,

if m+−m−1= p,

otherwise.

1(m+,m−;p) =

?m++m−

?

w′

1(m+,m−;p).

Lemma 11 For every k,l ∈ N, ? p = (p1,p2) ∈ Z2, and m+,m−∈ 2N,

w′

(m+−2k)!(m−−2l)!

2(k,l;m+,m−;? p)

?m+/2

k

?−1?m−/2

l

?−1

=

∑

2=m+,

1,m+

m+

1+m+

m+

2≥k

∑

2=m−,

1,m−

m−

m−

1+m−

2≥l

w′

1−k)!(m−

1(m+

1,m−

1;p1)

1−l)!·

(m+

w′

2−k)!(m−

1(m+

2,m−

2;p2)

2−l)!.

(m+

Moreover, the same identity holds when the w′

d’s are replaced by w′′

d’s respectively.

Proof: Map each walk w in W′

steps in dimension1 and 2 respectively. Let wi∈W′

each pair (w1,w2) is the image through the aforementioned mapping of

2(K,L;m+,m−;? p) to a pair of 1-dimensional walks w1and w2corresponding to the

1(m+

i,m−

i;pi), m+=m+

1+m+

2, and m−=m−

1+m−

2. We claim that

?m+−2|K|

m+

1−|K|

??m−−2|L|

m−

1−|L|

?

.

Page 15

elements of W′

in W′

Similarly, for v ∈ L we have that bv1 = 1 and bv2 = 2. When both of these conditions are satisfied, w is a preimage of

(w1,w2) if and only if among {aui : u ?∈ K,i = 1,2} and {bvi : v ?∈ L,i = 1,2} there are exactly m+

elements taking the value 1, respectively. Hence, there are?m+−2|K|

Fix K∗⊆U and L∗⊆V, such that |K∗| = k and |L∗| = l. By definition of w′

?m+/2

Observingthat??W′

w′

kl

m+

2(K,L;m+,m−;? p) provided m+

2(K,L;m+,m−;? p) be a preimage of the pair (w1,w2). Clearly, for u ∈ K it must hold that au1 = 1 and au2 = 2.

1,m+

2≥ |K| and m−

1,m−

2≥ |L|. Indeed, let w = au1

1···au2

m+bv1

1···bv2

m−

1−|K| and m−

1−|L|

m+

1−|K|

??m−−2|L|

m−

1−|L|

?possible choices for w. This completes

2(k,l;m+,m−;? p), we have

the proof of the claim.

w′

2(k,l;m+,m−;? p) =

k

??m−/2

1(m+

1,m−

?m+−2k

l

???W′

1;p1)·w′

2≥ |L|,

??m−−2l

2(K∗,L∗;m+,m−;? p)??.

1(m+

1(m+

1,m−

1;p1)×W′

1(m+

1,m+

?−1?m−/2

2,m−

2;p2)??=w′

?−1

1,m−

2,m−

2;p2), theaforementionedclaimand

the preceding equality imply that for m+

2≥ |K| and m−

2(k,l;m+,m−;? p)

?m+/2

=

1−km−

1−l

?

w′

1(m+

1,m−

1;p1)·w′

1(m+

2,m−

2;p2).

Hence,

w′

2(k,l;m+,m−;? p)

(m+−2k)!(m−−2l)!

?m+/2

k

?−1?m−/2

l

?−1

=

∑

2=m+,

1,m+

m+

1+m+

m+

2≥k

∑

2=m−,

1,m−

m−

m−

1+m−

2≥l

w′

1−k)!(m−

1(m+

1,m−

1;p1)·w′

1−l)!(m+

1(m+

2−k)!(m−

2,m−

2;p2)

2−l)!.

(m+

This proofs the equality involving the w′

d’s. The identity for the w′′

d’s can be established by a similar argument.

Corollary 2 For every ? p = (p1,p2) ∈ Z2and m+,m−∈ 2N,

w′

m+!m−!

(m+/2)k(m−/2)l

(m+)2k(m−)2l

2(m+,m−;? p)

=

∑

k≥0,l≥0

(−1)k+l

k!l!

·

∑

2=m+,

1,m+

m+

1+m+

m+

2≥k

∑

2=m−,

1,m−

m−

m−

1+m−

2≥l

w′

1−k)!(m−

1(m+

1,m−

1;p1)

1−l)!·

(m+

w′

2−k)!(m−

1(m+

2,m−

2;p2)

2−l)!.

(m+

Moreover, the same identity holds when the w′

d’s are replaced by w′′

d’s respectively.

Proof: By Lemma 10,

w′

2(k,l;m+,m−;? p)

m+!m−!

=(m+/2)k(m−/2)l

(m+)2k(m−)2l

∑

k,l≥0

(−1)k+l

k!l!

·

w′

2(k,l;m+,m−;? p)

(m+−2k)!(m−−2l)!

?m+/2

k

?−1?m−/2

l

?−1

.

Applying Lemma 11 yields the desired result.

The right hand side of the expression in Corollary 2 can be written in terms of primitives and derivatives of more

complicated expressions. This will allow us to derive an elegant formula for the left hand side expression. Let ∂x

denote the partial derivative with respect to x operator. The t times successive application of ∂x will be denoted ∂tx.

Similarly, let ∂−1x denote the inverse operator of ∂x (in other words, the operator returning a primitive with respect to

x). Also, we denote by ∂−tx the t times repeated application of ∂−1x.

Corollary 3 For i ∈ {1,2} and p ∈ Z, let

Ak,l(p) =

∑

s+≥k,s−≥l

w′

1(s+,s−;p)

(s+−k)!(s−−l)!(y1/2

+x)s+(y1/2

−x)s−.

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