---
title: Hello, J! Pascal's Triangle
route: /j-pascal
date: 2018-02-10
description:
---
Following [my previous post on the Fibonacci numbers](/j-fibonacci), here's a quick post on generating another famous mathematical sequence - [Pascal's Triangle](https://en.wikipedia.org/wiki/Pascal%27s_triangle).
```
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
```
## Quick Background
Pascal's triangle contains numbers formed by binomial coefficients (often referred to as "N choose K"), but there's a simple and elegant way to generate it: **each item is the sum of the two numbers above it**.
1 3 3 1
1 4 6 4 1
We see here that the underlined 6 is formed by adding the two threes above it.
1 3 3 1
1 4 6 4 1
Similarly, we see the 4 is formed by adding the 1 and 3, and the 1 is formed by adding 1 to nothing (0).
## Summing pairs
Before we get started go ahead and [grab yourself a copy of the J runtime](http://code.jsoftware.com/wiki/System/Installation/All-in-One) if you haven't already.
Let's start by grabbing the first 5 integers using the ["integers" verb `i.`](http://www.jsoftware.com/help/dictionary/didot.htm)
```
i. 5
0 1 2 3 4
```
We can use the ["same" verb `]`](http://www.jsoftware.com/help/dictionary/d500.htm) to echo whatever we pass into it.
```
] 10
10
] i.5
0 1 2 3 4
```
Our primary tool will be J's [infix adverb `\`](http://www.jsoftware.com/help/dictionary/d430.htm). This single backslash allows us to scan through a list of numbers in groups and apply a function to them.
Let's scan every 3 items and pass them to the "same" verb `]`.
```
3 ]\ i.5
0 1 2
1 2 3
2 3 4
```
We can see here that J looks through `0 1 2 3 4` in overlapping groups of 3. So we'll have `0 1 2`, followed by `1 2 3`, etc.
By changing the number in front, we can scan in groups of 2.
```
2 ]\ i.5
0 1
1 2
2 3
3 4
```
Let's swap out our boring `]` for something more interesting: `+/`, which sums a list of numbers.
```
3 +/\ i.5
3 6 9
+/ 0 1 2
3
+/ 1 2 3
6
+/ 2 3 4
9
```
Similarly, we can do this pairwise.
```
2 +/\ i.5
1 3 5 7
```
Pretty nifty right?
## Row after row
So why is this useful? Well, if we take another look at our triangle:
```
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
```
We can see that the next row, `1 5 10 10 5 1` can be formed by this exact strategy: summing each pair of numbers to generate the number beneath it.
```
2 +/\ 1 3 3 1
4 6 4
2 +/\ 1 4 6 4 1
5 10 10 5
```
Well, almost. Our issue is that we're missing out on the 1s on either side! This is because our infix operator `\` starts with `+/ 1 4` to get 5 - skipping over the 1.
To fix this, let's surround our row with 0s.
```
2 ]\ 0 1 3 3 1 0
0 1
1 3
3 3
3 1
1 0
2 +/\ 0 1 3 3 1 0
1 4 6 4 1
2 +/\ 0 1 4 6 4 1 0
1 5 10 10 5 1
```
Much better. So how do we surround with 0s automatically?
The [append verb `,`](http://www.jsoftware.com/help/dictionary/d320.htm) is used to join items and lists.
```
0 , 1
0 1
1 2 3 , 7 8 9
1 2 3 7 8 9
```
The ["bond" verb `&`](http://www.jsoftware.com/help/dictionary/d630n.htm) allows us to partially evaluate a verb that normally expects items on both sides.
```
1 + 2
3
(1&+) 2
3
4 % 2 NB. "%" is division!
2
(%&2) 4
2
```
We can bind our append verb to add a 0 to either side.
```
(0&,) 5 6 7
0 5 6 7
(,&0) 5 6 7
5 6 7 0
```
We can also do both.
```
(0&,) (,&0) 5 6 7
0 5 6 7 0
(,&0) (0&,) 5 6 7
0 5 6 7 0
```
Putting it all together: we can append 0 on either side of our row:
```
(0&,) (,&0) 1 4 6 4 1
0 1 4 6 4 1 0
```
And sum each pair:
```
2 +/\ (0&,) (,&0) 1 4 6 4 1
1 5 10 10 5 1
```
## All together now
Let's wrap this in our own verb called `next`. "`y`" gives us access to the argument(s) passed in.
```
next =: verb : '2 +/\ (0&,) (,&0) y'
next 1 3 3 1
1 4 6 4 1
next 1 4 6 4 1
1 5 10 10 5 1
```
Fortunately, `next` works just fine on the first row of our triangle: a single `1`.
```
next 1
1 1
```
We can apply `next` multiple times.
```
next 1
1 1
next 1 1
1 2 1
next next 1
1 2 1
next next next next 1
1 4 6 4 1
```
We can use the ["power" verb `^:`](http://www.jsoftware.com/help/dictionary/d202n.htm) to do this for us.
```
(next^:0) 1
1
(next^:1) 1
1 1
(next^:5) 1
1 5 10 10 5 1
```
We can also pass a list to `^:` to generate multiple powers at the same time.
```
(next^:(i.7)) 1
1 0 0 0 0 0 0
1 1 0 0 0 0 0
1 2 1 0 0 0 0
1 3 3 1 0 0 0
1 4 6 4 1 0 0
1 5 10 10 5 1 0
1 6 15 20 15 6 1
```
We can store this as another verb to generate Pascal's Triangle. We’ll also simplify our definition of next using the ["atop" verb `@`](http://www.jsoftware.com/help/dictionary/d620.htm).
```
next =: 2 +/\ (0&,) @ (,&0)
pascal =: verb : '(next^:(i.y)) 1'
NB. We can also inline `next` entirely
pascal =: verb : '((2 +/\ (0&,) @ (,&0))^:(i.y)) 1'
pascal 10
1 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0
1 2 1 0 0 0 0 0 0 0
1 3 3 1 0 0 0 0 0 0
1 4 6 4 1 0 0 0 0 0
1 5 10 10 5 1 0 0 0 0
1 6 15 20 15 6 1 0 0 0
1 7 21 35 35 21 7 1 0 0
1 8 28 56 70 56 28 8 1 0
1 9 36 84 126 126 84 36 9 1
```
🎉 🎉
Thanks to the powers of `\` and `^:`, we can intuitively generate Pascal's Triangle in a few dozen characters.
J makes operations on lists and matrices a breeze, and contains a number of interesting ideas for function composition and application. I hope you found this post interesting and give a look at what J has to offer.
I think you'll be pleasantly surprised at how fun it is.
Thanks for reading!