# ETOOBUSY ðŸš€ minimal blogging for the impatient

# PWC108 - Bell Numbers

**TL;DR**

On with TASK #2 from the Perl Weekly Challenge #108. Enjoy!

# The challenge

Write a script to display top 10

`Bell Numbers`

. Please refer to wikipedia page for more information.

# The questions

A fundamental question in this challenge is *what do you mean by
display?*

I mean, displaying a *number* is printing it, *right*? Or should we
actually *display all possible arrangements leading to that number?*.

# The solution

Weâ€™ll take the easy route in this, but still keep the door open for
possible future wast**AHEM**improvements.

As it is so often, the wikipedia page provides a neat way to calculate the numbers iteratively. This implies a time complexity of $O(n^2)$ and a space complexity of $O(n)$, which is not bad considering that weâ€™re asked to address only a handful of numbers and that the time complexity actually accounts for computing all $n$ numbers (so, on average, the complexity of any single of them is $O(n)$, assuming we can rely on the previous calculations).

Enough talking, letâ€™s get to the code:

```
#!/usr/bin/env perl
use 5.024;
use warnings;
use experimental qw< postderef signatures >;
no warnings qw< experimental::postderef experimental::signatures >;
sub bell_number ($N) {
state $cache = [1];
state $line = [1];
while ($cache->$#* < $N) {
my @previous_line = $line->@*;
$line->@* = $previous_line[-1];
push $line->@*, $_ + $line->[-1] for @previous_line;
push $cache->@*, $line->[0];
}
return $cache->[$N];
}
printf "B%d: %d\n", $_, bell_number($_) for 0 .. 9;
```

We keep two cache variables: one for the sequence itself (so that we can
reuse results if we need), one for the *previous line* in the triangle
to calculate the numbers. If you donâ€™t know what triangle Iâ€™m talking
about, look at the Triangle scheme for calculations.

I guess this is all *for this post* ðŸ˜…

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