Binomial Coefficients and Pascal’s Triangle

2015 ◽  
pp. 87-98
Keyword(s):  
Binomial Coefficients ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

scholarly journals Pascal type properties of Betti numbers

1994 ◽  
Vol 17 (3) ◽  
pp. 545-552
Author(s):  
Tilak de Alwis
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Binomial Coefficients ◽  
Minimal Resolution ◽  
Pascal’S Triangle ◽  
Specific Formula ◽  
Pascal's Triangle

In this paper, we will describe the Pascal Type properties of Betti numbers of ideals associated ton-gons. These are quite similar to the properties enjoyed by the Pascal's Triangle, concerning the binomial coefficients. By definition, the Betti numbersβt(n)of an idealIassociated to ann-gon are the ranks of the modules in a free minimal resolution of theR-moduleR/I, whereRis the polynomial ringk[x1,x2,…,xn]. Herekis any field andx1,x2,…,xnare indeterminates. We will prove those properties using a specific formula for the Betti numbers.

3. Permutations and combinations

2016 ◽  
pp. 28-49
Author(s):  
Robin Wilson
Keyword(s):  
Binomial Coefficients ◽  
Combination Rules ◽  
Binomial Theorem ◽  
Breakfast Cereals ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

Permutations and combinations have been studied for thousands of years. ‘Permutations and combinations’ considers selecting objects from a collection, either in a particular order (such as when ranking breakfast cereals) or without concern for order (such as when dealing out a bridge hand). It describes and investigates four types of selection—ordered selections with repetition, ordered selections without repetition, unordered selections without repetition, and unordered selections with repetition—and shows how they are related to permutations, combinations, the three combination rules, factorials, Pascal’s triangle, the binomial theorem, binomial coefficients, and distributions.

More power to Pascal

2008 ◽  
Vol 92 (525) ◽  
pp. 454-465 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Fibonacci Numbers ◽  
Ancient Greece ◽  
Binomial Coefficients ◽  
Medieval Italy ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

Pascal’s triangle is the most famous of all number arrays - full of patterns and surprises. One surprise is the fact that lurking amongst these binomial coefficients are the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers which arose in the Hindu studies of arrangements and selections, together with the Fibonacci numbers from medieval Italy. New identities continue to be discovered, so much so that their publication frequently excites no one but the discoverer.

Subprime Factorization and the Numbers of Binomial Coefficients Exactly Divided by Powers of a Prime

Integers ◽  
2012 ◽  
Vol 12 (2) ◽  
Author(s):  
William B. Everett
Keyword(s):  
Recurrence Relations ◽  
Binomial Coefficients ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

Abstract.We use the notion of subprime factorization to establish recurrence relations for the number of binomial coefficients in a given row of Pascal's triangle that are divisible by

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.

Pascal-like triangles and Fibonacci-like sequences

2010 ◽  
Vol 94 (529) ◽  
pp. 27-41
Author(s):  
H. Matsui ◽  
D. Minematsu ◽  
T. Yamauchi ◽  
R. Miy Adera
Keyword(s):  
Fibonacci Sequence ◽  
Binomial Coefficients ◽  
Extended Version ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

In [1] and [2] we demonstrated how Pascal-like triangles arose from the probabilities associated with the various outcomes of a particular game (see Definition 1 below). It was also shown that they could be considered as generalisations of Pascal's triangle. In this article we show how Fibonacci-like sequences arise from our Pascal-like triangles, and demonstrate the existence of simple relationships between these Fibonacci-like sequences and the Fibonacci sequence itself. In addition we will investigate a generalisation of the binomial coefficients that appears when considering an extended version of the game. We start by describing this game.

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.

Binomial Coefficients and The Partitioning of N-Dimensional Space

1978 ◽  
Vol 71 (8) ◽  
pp. 698-701
Author(s):  
Richard Crouse
Keyword(s):  
Dimensional Space ◽  
Binomial Coefficients ◽  
Pascal’S Triangle ◽  
Higher Dimensional ◽  
Pascal's Triangle

An example of extending Pascal's triangle to “higher” dimensional applications.

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