Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0015
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter deals with Λsubscript Greek capital letter gamma and Λsubscript Greek capital letter delta, and Statement G. It begins by introducing a nodal signature, denoted by η, and a ...
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This chapter deals with Λsubscript Greek capital letter gamma and Λsubscript Greek capital letter delta, and Statement G. It begins by introducing a nodal signature, denoted by η, and a subsignature, denoted by σ. It then presents the proof, using the signature σ to determine the rules for boxing and circling in α. While the circling in the accordion strictly speaking occurs at αᵢ and βsubscript i plus 1, we may equivalently consider it to occur at αᵢ and βᵢ for bookkeeping purposes. The chapter also considers the condition when the alternating sum for Λsubscript Greek capital letter gamma will only contain nonzero contributions from subsignatures.Less
This chapter deals with Λsubscript Greek capital letter gamma and Λsubscript Greek capital letter delta, and Statement G. It begins by introducing a nodal signature, denoted by η, and a subsignature, denoted by σ. It then presents the proof, using the signature σ to determine the rules for boxing and circling in α. While the circling in the accordion strictly speaking occurs at αᵢ and βsubscript i plus 1, we may equivalently consider it to occur at αᵢ and βᵢ for bookkeeping purposes. The chapter also considers the condition when the alternating sum for Λsubscript Greek capital letter gamma will only contain nonzero contributions from subsignatures.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0016
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter presents purely combinatorial results that are needed for the proof of Statement G. The motivation for these results comes from the appearance of divisibility conditions through the ...
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This chapter presents purely combinatorial results that are needed for the proof of Statement G. The motivation for these results comes from the appearance of divisibility conditions through the factor δn(Σ; α) defined in (15.2) that appears in Theorems 15.3 and 15.4. According to Statement F, the sum of Λsubscript Greek capital letter gamma(α, σ) over an f-packet is equal to the corresponding sum of ΛΔ(α′, σ). In order to prove Statement F, the chapter proceeds by identifying terms in the resulting double sum that can be matched. It considers subsignatures of η and concurrence as an equivalence relation.Less
This chapter presents purely combinatorial results that are needed for the proof of Statement G. The motivation for these results comes from the appearance of divisibility conditions through the factor δn(Σ; α) defined in (15.2) that appears in Theorems 15.3 and 15.4. According to Statement F, the sum of Λsubscript Greek capital letter gamma(α, σ) over an f-packet is equal to the corresponding sum of ΛΔ(α′, σ). In order to prove Statement F, the chapter proceeds by identifying terms in the resulting double sum that can be matched. It considers subsignatures of η and concurrence as an equivalence relation.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0017
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter presents the tools to prove that the proof of Theorem 1.2 is reduced to Statement G. It begins with the lemma stating that the cardinality of each Γ-pack or Δ-pack is a power of 2. It ...
More
This chapter presents the tools to prove that the proof of Theorem 1.2 is reduced to Statement G. It begins with the lemma stating that the cardinality of each Γ-pack or Δ-pack is a power of 2. It then introduces a proposition in which (σ, Σ) is an origin for a Γ-equivalence class. It is easy to see that a Γ-swap does not change α, Γ(σ), while it decreases κΓ(σ) by 1. To finish the proof of the theorem, the bijection is constructed, where the number of divisibility conditions is necessarily constant when the bijection obtains.Less
This chapter presents the tools to prove that the proof of Theorem 1.2 is reduced to Statement G. It begins with the lemma stating that the cardinality of each Γ-pack or Δ-pack is a power of 2. It then introduces a proposition in which (σ, Σ) is an origin for a Γ-equivalence class. It is easy to see that a Γ-swap does not change α, Γ(σ), while it decreases κΓ(σ) by 1. To finish the proof of the theorem, the bijection is constructed, where the number of divisibility conditions is necessarily constant when the bijection obtains.
