### Abstract

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.

Language | English |
---|---|

Article number | P1.45 |

Number of pages | 24 |

Journal | The Electronic Journal of Combinatorics |

Volume | 23 |

Issue number | 1 |

Publication status | Published - 4 Mar 2016 |

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### Keywords

- web diagram
- black diamond product
- idempotence
- combinatorial identity
- web world

### Cite this

*The Electronic Journal of Combinatorics*,

*23*(1), [P1.45].

}

*The Electronic Journal of Combinatorics*, vol. 23, no. 1, P1.45.

**Web matrices : structural properties and generating combinatorial identities.** / Dukes, Mark; White, Chris D.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Web matrices

T2 - The Electronic Journal of Combinatorics

AU - Dukes, Mark

AU - White, Chris D

PY - 2016/3/4

Y1 - 2016/3/4

N2 - 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.

AB - 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.

KW - web diagram

KW - black diamond product

KW - idempotence

KW - combinatorial identity

KW - web world

UR - http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p45

M3 - Article

VL - 23

JO - The Electronic Journal of Combinatorics

JF - The Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

M1 - P1.45

ER -