Web matrices: structural properties and generating combinatorial identities

In this paper we present new results for the combinatorics of web diagrams and web worlds. These are discrete objects that arise in the physics of calculating scattering amplitudes in non-abelian gauge theories. Web-colouring and web-mixing matrices (collectively known as web matrices) are indexed by ordered pairs of web-diagrams and contain information relating the number of colourings of the first web diagram that will produce the second diagram. We introduce the black diamond product on power series and show how it determines the web-colouring matrix of disjoint web worlds. Furthermore, we show that combining known physical results with the black diamond product gives a new technique for generating combinatorial identities. Due to the complicated action of the product on power series, the resulting identities appear highly non-trivial. We present two results to explain repeated entries that appear in the web matrices. The first of these shows how diagonal web matrix entries will be the same if the comparability graphs of their associated decomposition posets are the same. The second result concerns general repeated entries in conjunction with a flipping operation on web diagrams. We present a combinatorial proof of idempotency of the web-mixing matrices, previously established using physical arguments only. We also show how the entries of the square of the web-colouring matrix can be achieved by a linear transformation that maps the standard basis for formal power series in one variable to a sequence of polynomials. We look at one parameterized web world that is related to indecomposable permutations and show how determining the web-colouring matrix entries in this case is equivalent to a combinatorics on words problem.


Introduction
Web diagrams are discrete objects that are subject to certain colouring and reconstruction operations, and arise in physics in the calculation of scattering amplitudes in non-abelian gauge theories [9,16,20]. A prominent example is the theory of quarks and gluons, Quantum Chromodynamics (QCD), which is of great topical relevance given its application to current experiments such as the Large Hadron Collider.
Whilst some properties of web-mixing matrices have been established from a physics point of view [9,12], a fully general understanding of their structure and properties remains elusive. Webcolouring and web-mixing matrices admit a purely combinatorial definition and this provides an alternative framework to the physical picture to elucidate their properties. Our ultimate hope is that a detailed understanding of web diagrams and their matrices can be used to dramatically improve the precision of theoretical predictions for particle collider experiments.
The combinatorics of web diagrams and web matrices is an interesting study in its own right in that it combines parts of order theory, graph theory, and the theory of permutations in a new and novel way. Our seminal paper on the combinatorics of web diagrams [5] looked at these objects from several angles and has a companion physics paper [4] which explains how the results are pertinent to particle physics applications 1 . We were able to show, for example, that for particular web diagrams, 2010 Mathematics Subject Classification. 05A05, 05A19, 05C50. 1 For a recent introduction to the latter, see ref. [22]. the diagonal entries of the web matrices corresponding to them depend on the generating function of the descent statistic formed by summing over the Jordan-Hölder set of all linear extensions of a web diagram's decomposition poset (partially ordered set). We examined some special (parameterized) classes of web worlds which could be 'exactly' solved, and one of these saw the introduction of a new permutation statistic that did not seem to have been previously studied. Another result is that the number of different diagrams in a web world can be given by a hook-length style formula on its representation matrix.
Our aim in this study is not just to build on some of the results and directions that were initiated in [5], but to investigate some new aspects of these web diagrams that have not been explored before. Amongst these, we will give a combinatorial proof of idempotency of the web-mixing matrices that was originally established by way of a physics argument in Gardi and White [12, §3]. We will also introduce the black diamond product on power series and show how the web-colouring matrix of a web world whose web diagrams can be 'partitioned' (in a sense to be made precise later) depend on this black diamond product. Furthermore, we will show how a physics argument gives rise to a new technique for generating combinatorial identities using the black diamond product. We think that this technique is interesting in that it has the potential to generate quite unusual looking identities due to the black diamond product behaving in quite a complicated way on the power series in question.
We will concern ourself with the web diagrams that were defined in [5] 2 . Let P = {p 1 , . . . , p n } be a set of n pegs that are rooted to a plane. A web diagram D on P is a set of 4-tuples e i = (a i , b i , c i , d i ) where the e i represents an edge between peg p a i and peg p b i for which the endpoint of e i on peg p a i is the c i th highest vertex on that peg from the plane, and the endpoint of e i on peg b i is the d i th highest vertex on that peg from the plane. The labels of the endpoints of edges on each peg must be distinct, and this is taken care of in the formal definition, Definition 1. We will sometimes abuse the terminology by referring to 'peg i' in place of 'peg p i '.
Two illustrations of web diagrams are given in Figure 1. The web world of a web diagram D is the set of all web diagrams that result from permuting the order in which vertices appear on a peg, and we usually denote it by W (D). The two web diagrams in Figure 1 are in the same web world. In the physics context in which such diagrams arise, the pegs represent quarks or gluons emanating from the interaction point, and the lines joining pegs correspond to additional gluons being radiated and absorbed. The conventional depiction of each web diagram is a so-called Feynman diagram, an example of which is shown on the left of Figure 2.
To every web world W we associate two important matrices, M (W ) and R (W ) , called the webcolouring and web-mixing matrices, respectively. The entries of these matrices are indexed by ordered pairs of web diagrams, and contain information about colourings of the edges of the first web diagram that yield the second diagram under a construction determined by a colouring.
In this paper we deepen our study of these recently defined structures, with the goal of ascertaining general properties. Section 2 contains some background and definitions necessary for the rest of the paper. In Section 3 we introduce the black diamond product on formal power series and prove a decomposition theorem for the web matrices of web worlds that may be partitioned into web worlds on disjoint peg sets. In Section 4 we give a new method for generating combinatorial identities by exploiting some physical properties of web worlds.
In Section 5 we give two new results to explain the repeated entries that one notices occurring in web matrices. The first of theses results concerns diagonal entries and shows that if the comparability graphs of the decomposition posets of two web diagrams (that satisfy a further technical condition) are the same, then the diagonal entries on the web matrices for these two diagrams will be the same. The second result explains how the action of flipping a web diagram upside-down 2 The diagrams we consider here are referred to as Multiple Gluon Exchange Webs (MGEWs) in refs. [8,10]. Other types of web are possible, but are beyond the scope of the present work.  Figure 1. Two examples of web diagrams in the same web world. Web diagrams (a) and (b) in this figure both have a single edge between 'pegs' p 2 and p 4 . However the heights of the endpoints differ in both diagrams. In (a) the heights of this edge on pegs p 2 and p 4 are, respectively, 1 and 2. We represent this edge by the 4-tuple (2,4,1,2). In (b) the heights of this edge are, respectively, 3 and 1. The edge in this diagram is represented by the 4-tuple (2,4,3,1). The web graph of the web world (to be defined in Section 2) containing the web diagrams is shown as (c).
combines with the colouring and reconstruction operations on that same web diagram. In Section 6 we derive an expression for the entries of the square of a web-colouring matrix in terms of the entries of the web-colouring matrix, and give a combinatorial proof that the web-mixing matrices are idempotent. Finally, in Section 7 we investigate a particular web world whose diagrams consist of only two pegs that have multiple edges between them. We show how they are related to indecomposable permutations and how calculating the entries of the web matrices for this web world reduces to a combinatorics on words problem. Although web diagrams were defined in Section 1, we give here the formal specification in order to remove any uncertainty surrounding their definition. Definition 1. A web diagram on n pegs having m edges is a collection D = {e j = (a j , b j , c j , d j ) : 1 ≤ j ≤ m} of 4-tuples that satisfy the following properties:

Definitions and terminology
(ii) For i ∈ [1, n] let p i (D) be the number of j such that a j or b j equals i, that is, the number of edges in D incident with peg i. Then the labels of the vertices on peg i must be the first p i (D) positive integers, i.e.
and no vertex on a peg can be both a left endpoint and a right endpoint of an edge, i.e. A web diagram D may be mapped to a simple graph G( [1, n], E) where E(G) = {{a, b} : (a, b, c, d) ∈ D}. Web worlds are in 1-1 correspondence with loop-free simple graphs whose edges have labels in the set of positive integers. We call the labeled graph associated with a web world its web graph. The heights of the endpoints correspond to their distance from the meeting points of the the 4 arrowed lines. The 'pegs and plane' representation is shown to its right.
The vertices of the web graph correspond to the pegs of the web diagram (all vertices on a peg in the web diagram are projected to a single vertex in the web graph). The edges of the web graph are labeled with the multiplicities of edges between the pegs in the web diagram. The web world that the web diagrams (a) and (b) of Figure 1 are members of is represented by the web graph (c).
If there exist two non-empty web diagrams E and F such that D = E ⊕ F then we say that D is decomposable. Otherwise we say that D is indecomposable.
A k-colouring α of a web diagram D is an assignment of the numbers {1, . . . , k} to the edges of D such that the mapping α : D → {1, . . . , k} is surjective. In this instance we write |α| = k. Let Colours k (W ) be the set of k-colourings of web diagrams in W = W (D), and Colours(W ) = The construction procedure that we alluded to in the Introduction happens in the following way: a ℓ-colouring α of a web diagram D produces ℓ new sub-web diagrams D α (1), . . . , D α (ℓ), each of which consists of only the edges having that designated colour. These ℓ diagrams are not yet web diagrams, but we can relabel the heights of the vertices on each of the pegs to make them into web diagrams. Let rel be this relabelling operation. The new web diagram D ′ = Recon(D, α) = rel(D α (1)) ⊕ · · · ⊕ rel(D α (ℓ)) is formed by stacking the diagrams on top of one another in increasing order of the colours. The (D 1 , D 2 ) entries of the web-colouring and web-mixing matrices of W are: As stated in the introduction, these web-mixing matrices occur in the calculation of scattering amplitudes in QCD. More specifically, they occur in the exponents of amplitudes, and dictate how the colour charge and kinematic degrees of freedom of highly energetic quarks and gluons are entangled by the radiation of additional lower energy gluons.
These are illustrated in Figure 3. There are three different colourings of D 1 : The physical relevance of this example is that, owing to the second row of this matrix having no non-zero entries, only diagram D 1 survives in the exponents of scattering amplitudes. Further examples of these definitions and details can be found in [5, §2].
Every web diagram D may be decomposed and written as a sum of indecomposable web diagrams. A poset (partially ordered set) P on the set of these constituent indecomposable web diagrams can be defined as follows: Define the relation < 1 on P × P as follows: Let < 2 be the transitive closure of < 1 on P × P and let be the reflexive closure of < 2 on P × P . We call P (D) = (P, ) the decomposition poset of D.
In the following sections we will repeatedly come across the Fubini polynomials (also known as the ordered Bell polynomials) F n (x) = n k=1 k! n k x k .

A decomposition theorem for web matrices
The main result of this section is a theorem that explains how to form the web-colouring matrix of a web world that admits a decomposition as a disjoint union of at least two other web worlds. Physical arguments tell us that such webs do not contribute to the exponents of scattering amplitudes [9] and this is something we will discuss further and exploit in Section 4.
In order to be able to state our main theorem, we must first define some new numbers and a new product on formal power series. Let k i 1 ,...,im ⋆ be the number of 0-1 fillings of an m row and k column array such that there are precisely i 1 ones on the top row, i 2 ones in the second row, etc., and there are no columns of only zeros.
Proof. The generating function for an entry in row j of the m × k array is u 0 j + u 1 j = 1 + u j . The only forbidden configuration in the array is a column of all zeros, so the generating function for columns is  Since there are no restrictions on rows, the generating function for the m × k array is therefore The number of arrays with i 1 ones in the first row, i 2 ones in the second row, and so on, is the coefficient of u i 1 1 u i 2 2 · · · u im m in the above generating function: It is a simple exercise to check that We will now define a new product of power series that will both help in presenting our results concerning web worlds, but also serves in establishing a new automatic way to generate general combinatorial identities from web worlds. The black diamond product is a binary operation that is both commutative and associative, and we define it in a very general form since this is how we will typically be using it.
Let us note that if the power series A (i) (x) are each polynomials of degree n i , then the product may be written in the more computationally efficient form: ( The unit of the black diamond product is 1, i.e. 1 A(x) = A(x). To recap on our point before the definition, due to commutativity and associativity the order in which power series appear in the black diamond product does not change its outcome. In other words, given the power series [1,m] and a permutation π ∈ S m , We will abbreviate where there are m ones on the right hand side. The value k 1,...,1 ⋆ is the number of ways to fill a table of k columns and m rows with 0s and 1s such that there is exactly one 1 in every row and there are no columns of only 0s. This is simply another way to encode an ordered set partition of an m-set into k sets (if a is in the ith set of such a sequence then the solitary 1 in row a of the array will be in the ith column). This number is k! m k and so for all m ≥ 1, Example 10. Using the fact that Several of the power series that arise from taking the black diamond product of simple power series expressions appear to correspond to known sequences/power series. This suggests the black diamond product may be an object worthy of study in its own right. Some examples of these include: x x n = nx n + (n + 1)x n+1 , the coefficients in the power series x n x n+1 appear to be given by [17, A253283], and the sequence of coefficients of x n x n = n k=0 n + k k n k x n+k is known to count several different structures (see [17, A063007]). (We omit the proof of this final observation since it is not immediately relevant to the paper's goal.) We are now ready to state our main theorem.
Proof. Suppose that α 1 is an i 1 -colouring of D 1 which constructs D ′ 1 (i.e. Recon(D 1 , α) = D ′ 1 ) and that α 2 is a i 2 -colouring of D 2 that constructs D ′ 2 , and so on. An i-chain is a totally ordered set of i elements. The number of ways to k-colour the diagram D = D 1 ⊕ · · · ⊕ D m so that it becomes D ′ is the number of ways to embed the m chains i 1 -chain, i 2 -chain, . . ., i m chain, which represent the ordered colourings of D 1 , . . . , D m , respectively, into a k-chain.
The value of k must be at least the length of the largest of the m chains, and of course must be at most the sum of the lengths of all constituent m chains, i.e. k ∈ [max(i 1 , . . . , i m ), i 1 + · · · + i m ]. Furthermore, this embedding must be surjective, for otherwise at least one of the k-colours {1, . . . , k} would not have any corresponding colour in the m constituent chains. One can recast this in the form of a tabular 0-1 filling problem where we have m rows and k columns whereby the column indices of the i r ones that appear in row r of the table indicate the new colours that they take on in the k-colouring. The surjectivity condition translates into there being no columns of all zeros. The number of ways to do this is therefore Suppose that the entries of the web-colouring matrices that correspond to the diagrams are i j ways to colour D j to produce D ′ j for all j ∈ [1, m], the factor a (1) im must be included. Therefore from Example 9.
Corollary 13. With the same setup as in Theorem 11, Dm,Dm (x).

A method for generating combinatorial identities
In this section we will outline a method for generating combinatorial identities by using the black diamond product in conjunction with a result concerning the trace of web-mixing matrices for disjoint web worlds.

Proposition 14.
Let W be a web world that is the disjoint union of at least two web worlds. Then all entries of the web-mixing matrix R (W ) are zero, and consequently trace R (W ) = 0.
A statistical physics-based proof of this result has been given in Gardi et al. [9, Section 5.1]. The essential physical idea, elaborated further in [11], is that exponents of scattering amplitudes can only contain interactions between non-disjoint groups of quarks and gluons.
The relationship between the web-mixing matrices and web-colouring matrices is Integrating formal power series can be a contentious issue, so let us be clear that the integrals we perform in this paper are definite integrals and always correspond to the transformation: Suppose that W 1 , . . . , W n are web worlds on disjoint peg sets that all have the same web-colouring matrix M = M (W i ) (x). (This happens for example when all web worlds are equivalent to one another by relabeling the peg sets that they are defined upon.) Suppose further that the diagonal entries of M are (G 1 (x), . . . , G t (x)) and that For the case that there are only two different power series that appear as diagonal entries of M , we have Theorem 15. Let W be a web world whose web-colouring matrix has s different diagonal entries (H 1 (x), . . . , H s (x)) that appear with multiplicities (h 1 , . . . , h s ). Then for all positive integers m, we have The expression for the s = 2 case is: We will apply the above theorem to two extremely simple web worlds to see the combinatorial identities that emerge.
This gives us the identity: This identity can also be found in [21,Eqn. 27].
Example 17. Let W be the web world of Example 4, with In this case s = 2, H 1 (x) = x, H 2 (x) = x + 2x 2 , and h 1 = h 2 = 1. From Equation 4, we have H 1 (x) m = F m (x). One can show that H 2 (x) m = F 2m (x) and then applying Theorem 15 we have This gives us the identity We do not know if this is a known identity.

Repeated entries in web matrices
One observation that is apparent when calculating the web matrices (be they web-mixing or web-colouring) of web worlds is that there are a large number of entries that are the same. A deeper understanding of this property would simplify the calculation of web-mixing matrices, and has even led, for certain families of web diagram, to obtaining the web-mixing matrix for arbitrary numbers of gluons [4]. In this section we prove two new results to explain some of these repetitive entries. The first theorem gives one explanation for repeated entries found on the diagonal of a web-colouring matrix, and also explains the same for the web-mixing matrix since the entries of the latter are a simple integral transformation of those in the web-colouring matrix. This theorem also builds on our earlier result [5,Thm. 3.4].
Given a poset P = (P, ≺), its comparability graph comp(P ) is the graph whose vertices are the elements of P , with x, y ∈ P adjacent in comp(P ) if x ≺ y or y ≺ x.
Theorem 18. Let D and D ′ be web diagrams in a web world W with D = E 1 ⊕ · · · ⊕ E k and D ′ = E ′ 1 ⊕ · · · ⊕ E ′ k ′ , where each of the constituent diagrams E i and E ′ i are indecomposable. Suppose that the members of (E 1 , . . . , E k ) are distinct and the members of (E ′ 1 , . . . , E ′ k ′ ) are also distinct. If comp(P (D)) = comp(P (D ′ )), then M Proof. First let us consider the web diagram D = E 1 ⊕ · · · ⊕ E k where each of the diagrams E i is indecomposable. All ℓ-colourings α such that Recon(D, α) = D must have the property that α(e) = α(e ′ ) whenever e, e ′ ∈ E i because E i is indecomposable. Thus α must be a surjective map from {E 1 , . . . , E k } to [1, ℓ]. Equivalently α surjectively maps the k elements of the decomposition poset P (D) to the total order on [1, ℓ]. Let us call the number of these surjective maps Θ(P (D), ℓ). This number is related to the number Ω(P (D), ℓ) of order preserving maps from the poset P (D) to [1, ℓ] via an inclusion-exclusion argument: The same is true for D ′ , so we have If comp(P (D)) = comp(P (D ′ )) then by Stanley [18,Cor. 4.4] we have that Ω(P (D)) = Ω(P (D ′ )), and consequently the expressions in Equations 10 and 11 are the same.
Since those edges of D that are coloured ℓ using the colouring β are precisely the same as the edges that are coloured k + 1 − ℓ using the colouring α, we have D β (ℓ) = D α (k + 1 − ℓ) and Applying flip to both sides gives Recon(flip(D), β) = flip(D ′ ), as was required.
Theorem 23. Let W be a web world whose diagrams each have n edges. Let D, D ′ ∈ W and suppose that M Proof. Let α ∈ Colours(W ). Given two colourings α, α ′ ∈ Colours(W ) there is a unique γ ∈ Colours(W ) such that Recon(D, γ) = Recon(Recon(D, α), β). In such an instance we write γ ≡ β •α. D,D ′ (x) = z 1 x + · · · + z n x n . Then where N j,k (i) is the number of pairs (β, α) of colourings in Colours j (W ) × Colours k (W ) such that β • α is equivalent to a i-colouring γ ∈ Colours i (W ). To see how a colouring of a colouring of a diagram can be turned into one colouring 3 , we consider the ordered pairs of colours. The diagram D is coloured with α and reconstructed to give the diagram D ′′ . The diagram D ′′ is coloured with β and reconstructed to give the diagram D ′ .
In other words, the subweb-diagram of Recon(D, α) whose edges e have colour β(e) = 1 will maintain their order relative to one another in D ′ and be the lowest. But amongst these edges, which will be the lowest w.r.t. the colouring α of D? It will be those edges e ∈ D that are coloured α(e) = 1, followed by all those edges e ∈ D with α(e) = 2, and so on. Thus the lexicographic order on all such pairs of colourings that exists corresponds to the total order for a single colouring. For example, if we have the pairs of colourings (β(e), α(e)) = (1, 3), (1, 5), (2, 1), (2, 2), (2,5), (3,4), then these correspond to the colours γ(e) = 1, 2, 3, 4, 5, 6, respectively.
The number N j,k (i) is therefore the number of j × k zero-one matrices that have exactly i ones and there are neither columns nor rows of only zeros. This number is known to be (see [3,2] or [15,Eqn. 120] Therefore x j+k As mentioned above, the following theorem has already been proven using a physics argument (see [12,Sect. 3]). Here we give the first combinatorial proof.
Theorem 24. Let W be a web world whose web diagrams each have n edges. The web-mixing matrix R (W ) is idempotent; (R (W ) ) 2 = R (W ) .
Proof. Using the same method of summation as we did at the start of the proof of Theorem 23, The inner sum is n i,j=1 We now need to prove that for 1 ≤ k ≤ n, Let G(n, k, a) = n b=1 (−1) b n b ab−1 k−1 so that F (n, k) = n a=1 (−1) a n a G(n, k, a). Now Since k ∈ [1, n], and j ≤ k, the Stirling number on the right hand side of the previous equation will always be zero for the values that we are summing over, and the only term that will contribute will be −0 j−1 . This term itself will always be zero, except for when j − 1 = 0. This gives Therefore F (n, k) = n a=1 (−1) a n a (−1) k = (−1) k ((1 − 1) n − 1) = (−1) k−1 . This allows us to write Equation 14 as (−1) k−1 /k which, in turn, means that Equation 13 is

Indecomposable permutations and a web world on two pegs having multiple edges
In this section we consider the web world W n whose web graph is G(W n ) = K 2 and the single edge is labelled with n. This web world consists of diagrams having two pegs and n edges connecting the n vertices on the first peg to n vertices on the second peg. From a physics point of view, such webs have been studied for a long time [13,7,19]. However, they have yet to be analysed using the language of web-colouring and mixing matrices and which we will now attend to.
Every diagram D ∈ W n is uniquely described as a set of 4-tuples D = {(1, 2, i, π i ) : π ∈ S n } and consequently there are n! diagrams in W n . We can abbreviate this by writing D = D π . Given a permutation π = π 1 . . . π n and permutations α ∈ S ℓ , β ∈ S n−ℓ , we say that π is the sum of the permutations α and β if π i = α i for all i ≤ ℓ and π i = ℓ + β i−ℓ for all i > ℓ. We write this as π = α + β. If a permutation π can be written as such a sum of smaller permutations, then we say that π is decomposable. Otherwise we say that π is indecomposable. Let Ind n be the set of indecomposable permutations of length n and let Ind = ∪ n≥1 Ind n . Every permutation π ∈ S n admits a unique representation as a sum of indecomposable permutations. If π = σ 1 + · · · + σ k where every σ i ∈ Ind, then D π = D σ 1 ⊕ · · · ⊕ D σ k and we will write indparts(π) for the number k of indecomposable permutations in the sum.

Problem 25. [Word reconstruction generating function]
Given a finite word w = w 1 . . . w n on a finite alphabet, let f w (k) be the number of ways to read the word from left to right k times (each time always picking some unchosen letter) so that we arrive back at the original word w. Let F w (x) = k f w (k)x k .

(W )
Dπ ,Dπ (x) = F w (x). Proof. Let D π ∈ W n with π = σ 1 ⊕ · · · ⊕ σ m . In order to colour the edges of D π so that the same diagram is reconstructed, the colours of the edges that are in the same indecomposable part must remain the same. (Otherwise we would be constructing a web diagram which has more indecomposable parts than the original one, which would be a contradiction.) Given a set A = {a 1 , . . . , a i }, define σ A = σ a 1 ⊕ · · · ⊕ σ a i where a 1 < · · · < a i . Let  The set of colourings c ′ of edges of D π which results in a reconstruction of the diagram is in 1-1 correspondence with the set of colourings c ∈ Colourings(m, ·) such that σ 1 ⊕ · · · ⊕ σ m = σ Colc(1) ⊕ · · · ⊕ σ Colc(k) . This is the word reconstruction problem (Problem 25) for the word w = (σ 1 , . . . , σ m ) which is on the alphabet {σ 1 , . . . , σ m }.
There is no simple formula for the diagonal entries of the web-colouring matrix for this web world due to there being no 'closed form' answer to Problem 25. However, we have noticed some potential structure from looking at the trace of the web-colouring matrices for small values of n, outlined in Figure 7. The numbers from Figure 7 support the following conjecture for all n ∈ [1,8].