path: root/hidden/2018-02-10-j-pascal.md

---
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**.

<pre>
 1 <span style="border:1px solid black">3 3</span> 1
1 4 <u>6</u> 4 1
</pre>

We see here that the underlined 6 is formed by adding the two threes above it.

<pre>
 <span style="border:1px solid black">1 3</span> 3 <span style="border:1px solid black">1  </span>
1 <u>4</u> 6 4 <u>1</u>
</pre>

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!