Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron Part I: Binomial Coefficients, Extended Binomial Coefficients and Preparation for Further Work

1999 ◽  
Vol 30 (3) ◽  
pp. 170-186
Author(s):  
Peter Hilton ◽  
Jean Pedersen
Keyword(s):  
Binomial Coefficients ◽  
Pascal Triangle ◽  
And Algebra

Preserving log-concavity for p,q-binomial coefficient

2019 ◽  
Vol 11 (02) ◽  
pp. 1950017
Author(s):  
Moussa Ahmia ◽  
Hacène Belbachir
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Linear Transformations ◽  
Pascal Triangle

We study the log-concavity of a sequence of [Formula: see text]-binomial coefficients located on a ray of the [Formula: see text]-Pascal triangle for certain directions, and we establish the preserving log-concavity of linear transformations associated to [Formula: see text]-Pascal triangle.

scholarly journals -DEFORMED RATIONALS AND -CONTINUED FRACTIONS

2020 ◽  
Vol 8 ◽  
Author(s):  
SOPHIE MORIER-GENOUD ◽  
VALENTIN OVSIENKO
Keyword(s):  
Continued Fractions ◽  
Jones Polynomial ◽  
Total Positivity ◽  
Recursive Formula ◽  
Rational Numbers ◽  
Binomial Coefficients ◽  
Indecomposable Representation ◽  
Pascal Triangle ◽  
Rational Knots ◽  
Gaussian Binomial Coefficients

We introduce a notion of $q$ -deformed rational numbers and $q$ -deformed continued fractions. A $q$ -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$ -deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$ -rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$ -deformation of the Farey graph, matrix presentations and $q$ -continuants are given, as well as a relation to the Jones polynomial of rational knots.

scholarly journals Generalized Pascal triangle for binomial coefficients of words

2016 ◽  
Vol 80 ◽  
pp. 24-47 ◽  
Author(s):  
Julien Leroy ◽  
Michel Rigo ◽  
Manon Stipulanti
Keyword(s):  
Binomial Coefficients ◽  
Pascal Triangle

Stop-sign theorems and binomial coefficients

2010 ◽  
Vol 94 (530) ◽  
pp. 247-261 ◽  
Author(s):  
Peter Hilton ◽  
Jean Pedersen
Keyword(s):  
Quality Of Life ◽  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Original Form ◽  
Pascal Triangle ◽  
Stop Sign

Dedicated to the memory of Russell Towle, a remarkable man who contributed so much to geometry and to other aspects of the quality of life.We introduce an expanded notation where r + s = n, for the binomial coefficient , and then use this expanded notation to develop theorems involving 8 binomial coefficients, analogous to the Star of David Theorem, which. in its original form, involved the 6 neighbours of a given binomial coefficient in the Pascal Triangle (see Section 3), that appeared in [1,2,3,4,5,6,7,8,9].

scholarly journals On Unimodality Problems in Pascal's Triangle

10.37236/837 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Xun-Tuan Su ◽  
Yi Wang
Keyword(s):  
Affirmative Answer ◽  
Binomial Coefficients ◽  
Pascal’S Triangle ◽  
Pascal Triangle ◽  
Pascal's Triangle

Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We also propose two more general conjectures.

Generalising Pascal's Triangle

2004 ◽  
Vol 88 (513) ◽  
pp. 447-456 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Prime Number ◽  
Recurrence Relation ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Combinatorial Properties ◽  
Pascal’S Triangle ◽  
Binomial Expansion ◽  
Pascal Triangle ◽  
Divisibility Property ◽  
Pascal's Triangle

Pascal's triangle, the Binomial expansion and the recurrence relation between its entries are all inextricably linked. In the normal course of events, the Binomial expansion leads to Pascal's Triangle, and thence to the recurrence relation between its entries. In this article we are going to reverse this process to make it possible to explore a particular type of generalisation of such interlinked structures, by generalising the recurrence relation and then exploring the resulting generalised ‘Pascal Triangle’ and ‘Binomial expansion’. Within the spectrum of generalisations considered, we find exactly four of particular significance: those concerned with the Binomial coefficients, the Stirling numbers of both kinds, and a lesser known set of numbers – the Lah numbers. We also examine the combinatorial properties of the entries in these triangles and a prime number divisibility property that they all share. Thereby, we achieve a remarkable synthesis of these different entities.

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