Dyalog#APLQuest 2015-10: Write a function that takes an integer right argument representing the order of the Pascal's triangle to be created and returns a “nicely” formatted character matrix (see https://apl.quest/2015/10/ to test your solution and view ours).
Fractal KittyNew post on Fractal Kitty:
https://www.fractalkitty.com/filtering-snowflakes/
#mathart #pascalsTriangle #binomialExpansion #mtbos #snowflakes #iteachmath
Filtering Snowflakes
Whether you call this triangle Pascal's triangle, Binomial Expansion Coefficients, Yang Hui's triangle, or any other name, it is beautiful. Finding patterns in this triangle is fun - from counting numbers, to looking at parity (even/odd-ness), to primes and other numbers. When we look for certain numbers, we can
Niles JohnsonI'm doing some symmetric monoidal algebra that involves careful counting of some signs of certain permutations. The work depends on a couple of combinatorial identities that I haven't seen anywhere else—do you recognize these (below)?!
The identities involve the "choose two" binomial coefficients.
For ease of typing, I'll use this notation:
[a;2] = binomial(a,2) = a·(a-1)/2
(read "a choose 2")
The two identities are
(I1):
[a+b;2] = [a;2] + [b;2] + ab
and
(I2):
[ab;2] = a[b;2] + b[a;2] + 2[a;2][b;2]
In particular, (I2) means there is a mod 2 congruence
[ab;2] ≡ a[b;2] + b[a;2]
and that's the form that has been particularly useful for me.
Neither of these identities are hard to prove directly from the definition, and they hold for positive *and negative* integers a and b. (That extension to all integers is important for my applications too.)
I've done some internet searching (wikipedia [1,2] and other general references), but I haven't found mention of these particular identities. So, I'm wondering if anyone here recognizes them. (Boosts appreciated!)
Note: These particular binomial coefficients [a;2], for positive a, are also called *triangular numbers*. I'll rewrite (I1) and (I2) in terms of triangular numbers in the next post, in case people will recognize that alternate form (but I doubt it).
[1] https://en.wikipedia.org/wiki/Binomial_coefficient
[2] https://en.wikipedia.org/wiki/Triangular_number
(1/2)
Lise Andreasen#ThisWeeksFiddler, 20251031
This week the #puzzle is: How Much Does Game 1 Matter? #probabilities #combinatorics #PascalsTriangle #animation You and your opponent are beginning a best-of-seven series, meaning the first team to win four games wins the series. Both teams are evenly matched, meaning each team has a 50 percent chance of winning each game, independent of the […]
https://stuff.ommadawn.dk/2025/11/04/thisweeksfiddler-20251031/
Knowledge Zone#DidYouKnow: In mathematics, #PascalsTriangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra.
Pascal's triangle determines the coefficients which arise in binomial expansions
bs2> We all know and love #PascalsTriangle and the #FibonacciSequence. But who knows #meruprasthara and the Pingala sequence?.. America invented #ModernDemocracy!.. Even though the #Haudenosaunee practiced it for centuries before on the same land. Let us be clear: we are very bad at attribution. Citing our sources is an echochamber, a way to further empower voices that are already empowered, whether they did the work and contributed the value or not.
https://drym.org/on-attribution/
#PingalaSequence
https://drym.org/on-attribution/
#PingalaSequence
PensionDanA more elegant path forward dawned on me (using #PascalsTriangle!), so I won't be using the equation in the preceding post.
First, some comments & definitions. Cycles will refer to cycles of length 2 or more (not including 1-cycles, which we will regard as positions along the main diagonal that don't get permuted). Define the delta of two consecutive elements of a cycle a, b as b - a. Clearly, the sum of all the deltas of a cycle is zero, as each element is added and subtracted once. Therefore, every cycle has at least one negative delta among its pairs of consecutive elements.
Define an increasing cycle as a cycle with just one negative delta. Observe that an increasing cycle can be written with its elements in increasing order (the delta from the last element back to the first element is the negative one).
Consider these sets of positions NEZ and SWZ (see illustration) in an n x n matrix, which lie in the antidiagonal and the sub-antidiagonal (thus in the eligible positions for (\Omega^{xx}_{n}\)):
• NEZ; the northeast zigzag of (n-1) positions that begins at the upper right corner, ordered as positions \(a_{1,n}, a_{2,n}, a_{2,n-1}, a_{3,n-1},…a_{[(n+1)/2],[(n+1)/2]+1}\).
• SWZ; the southwest zigzag of (n-1) positions that begins at the lower left corner, ordered as positions \(a_{n,1}, a_{n,2}, a_{n-1,2}, a_{n-1,3},…a_{[(n+1)/2]+1,[(n+1)/2]}\).
We’ll describe a set of cycles C, and then show that permutations which are a product of disjoint cycles from C are vertices of \(\Omega^{xx}_{n}\). As we know that \(\Omega^{xx}_{n}\) has \(2^{n-1}\) vertices, if we can verify that there are \(2^{n-1}\) such permutations, then we will have characterized all the vertices of \(\Omega^{xx}_{n}\).
First, some comments & definitions. Cycles will refer to cycles of length 2 or more (not including 1-cycles, which we will regard as positions along the main diagonal that don't get permuted). Define the delta of two consecutive elements of a cycle a, b as b - a. Clearly, the sum of all the deltas of a cycle is zero, as each element is added and subtracted once. Therefore, every cycle has at least one negative delta among its pairs of consecutive elements.
Define an increasing cycle as a cycle with just one negative delta. Observe that an increasing cycle can be written with its elements in increasing order (the delta from the last element back to the first element is the negative one).
Consider these sets of positions NEZ and SWZ (see illustration) in an n x n matrix, which lie in the antidiagonal and the sub-antidiagonal (thus in the eligible positions for (\Omega^{xx}_{n}\)):
• NEZ; the northeast zigzag of (n-1) positions that begins at the upper right corner, ordered as positions \(a_{1,n}, a_{2,n}, a_{2,n-1}, a_{3,n-1},…a_{[(n+1)/2],[(n+1)/2]+1}\).
• SWZ; the southwest zigzag of (n-1) positions that begins at the lower left corner, ordered as positions \(a_{n,1}, a_{n,2}, a_{n-1,2}, a_{n-1,3},…a_{[(n+1)/2]+1,[(n+1)/2]}\).
We’ll describe a set of cycles C, and then show that permutations which are a product of disjoint cycles from C are vertices of \(\Omega^{xx}_{n}\). As we know that \(\Omega^{xx}_{n}\) has \(2^{n-1}\) vertices, if we can verify that there are \(2^{n-1}\) such permutations, then we will have characterized all the vertices of \(\Omega^{xx}_{n}\).
#DidYouKnow: In mathematics, #PascalsTriangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra.
Pascal's triangle determines the coefficients which arise in binomial expansions
#DidYouKnow: In mathematics, #PascalsTriangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra.
Pascal's triangle determines the coefficients which arise in binomial expansions
R. Sunder-RajSome stuff that led to me thinking about a relationship between #PascalsTriangle and #binaryTrees.
It seems to have started with me trying to represent Pascal’s triangle using things like #crochet, #macrame, #braiding and #knitting.