scholarly journals Quiver Mutation Sequences and $q$-Binomial Identities

2017 ◽  
Vol 2018 (23) ◽  
pp. 7335-7358
Author(s):  
Akishi Kato ◽  
Yuma Mizuno ◽  
Yuji Terashima
Keyword(s):  
Quiver Mutation ◽  
Binomial Identities

scholarly journals TWO SUPERCONGRUENCES RELATED TO MULTIPLE HARMONIC SUMS

2021 ◽  
pp. 1-11
Author(s):  
ROBERTO TAURASO
Keyword(s):  
Binomial Identities

Abstract Let p be a prime and let x be a p-adic integer. We prove two supercongruences for truncated series of the form $$\begin{align*}\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}\end{align*}$$ which generalise previous results. We also establish q-analogues of two binomial identities.

scholarly journals Combinatorial and Automated Proofs of Certain Identities

10.37236/2010 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Justin Brereton ◽  
Amelia Farid ◽  
Maryam Karnib ◽  
Gary Marple ◽  
Alex Quenon ◽  
...  
Keyword(s):  
Modern Machine ◽  
Binomial Identities

This paper focuses on two binomial identities. The proofs illustrate the power and elegance in enumerative/algebraic combinatorial arguments, modern machine-assisted techniques of Wilf-Zeilberger and the classical tools of generatingfunctionology.

scholarly journals Coloured quiver mutation for higher cluster categories

2009 ◽  
Vol 222 (3) ◽  
pp. 971-995 ◽  
Author(s):  
Aslak Bakke Buan ◽  
Hugh Thomas
Keyword(s):  
Quiver Mutation ◽  
Cluster Categories

Binomial Identities

1946 ◽  
Vol 53 (1) ◽  
pp. 24
Author(s):  
R. E. Greenwood ◽  
A. M. Gleason
Keyword(s):  
Binomial Identities

scholarly journals The Existence of a Maximal Green Sequence is not Invariant under Quiver Mutation

10.37236/5412 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Greg Muller
Keyword(s):  
Maximal Green Sequence ◽  
Quiver Mutation

This note provides a quiver which does not admit a maximal green sequence, but which is mutation-equivalent to a quiver which does admit a maximal green sequence. The proof uses the `scattering diagrams' of Gross-Hacking-Keel-Kontsevich to show that a maximal green sequence for a quiver determines a maximal green sequence for any induced subquiver.

Undergraduate Students’ Combinatorial Proof of Binomial Identities

2021 ◽  
Vol 52 (5) ◽  
pp. 539-580
Author(s):  
Elise Lockwood ◽  
Zackery Reed ◽  
Sarah Erickson
Keyword(s):  
Undergraduate Students ◽  
Teaching Experiment ◽  
Combinatorial Proof ◽  
Representation System ◽  
Representation Systems ◽  
Symbolic Reasoning ◽  
Binomial Identities

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.

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