#pascalstriangle — Public Fediverse posts

New post on Fractal Kitty:

https://www.fractalkitty.com/filtering-snowflakes/

#mathart #pascalsTriangle #binomialExpansion #mtbos #snowflakes #iteachmath

I'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)

#binomial #triangular #PascalsTriangle

Some 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.

#mathart #mathsart #binaryTree

Some 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.

#mathart #mathsart #binaryTree

Some 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.

#mathart #mathsart #binaryTree

Some 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.

#mathart #mathsart #binaryTree

Some 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.

#mathart #mathsart #binaryTree

And another clip of a representation of a #PascalsTriangle #BinaryTree

#mathart

Here is a clip to do with the #PascalsTriangle #BinaryTree shown as individual paths layered together

#mathart

Splitting a #PascalsTriangle #BinaryTree into two copies.

#mathart

These ideas eventually worked their way into a #woven #BinaryTree representation of #PascalsTriangle.

#mathart