Talk:Pascal's triangle: Difference between revisions
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I think that maybe all example output should follow the task description format of an isosceles triangle. --[[User:Paddy3118|Paddy3118]] 08:59, 27 December 2009 (UTC)
:That's not always easy to do. I think the important part of the task is the generation of each row. We don't need to complicate it with output formatting that isn't important to the theory involved. --[[User:Mwn3d|Mwn3d]] 18:37, 27 December 2009 (UTC)
== J Explanation ==
<lang j>!/~@i. N</lang>
The triangle itself is simply the table of number-of-combinations, for the first N non-negative integers.
<lang J> !/~i.5
1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1</lang>
That is, [[wp:Combination|C(n,k)]] for all <tt>n,k in [0 .. n)</tt>. J's notation for <tt>C(n,k)</tt> is <code>k ! n</code> (mnemonic: combinations are closely related to factorials, which are denoted by <tt>!</tt> in math).
So, for example, the number of ways to choose a poker hand (5 cards from the deck of 52):
<lang j> 5!52
2598960</lang>
So <code>!</code> is the mathematical choose function. As for <code>/~@i.</code>, the <code>/~</code> would be "table of" and <code>i.</code> "the first N non-negative integers (i.e. 0 .. N-1)". (And <code>@</code> is just glue.)
But, formatting the thing takes a bit more effort:
<lang j> (-@|. |."_1 [: ;:inv [: ":@-.&0&.>@|: !/~)@i. 5
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1</lang>
Here, we have refactored the operation slightly:
<lang j>(stuff)@i.5</lang> makes the sequence 0 1 2 3 4 available every odd verb (counting from the right) in stuff
The "stuff" are (from right to left)
<lang j>!/~</lang> build our table of combinations
<lang j>":@-.&0&.>@|:</lang> transpose that array, and for each number remove it if it is zero, format it as a string and put it in a box
<lang j>[:</lang> placeholder meaning "no verb here" the verb to the right does not get a left argument
<lang j>;:inv</lang> combine the strings in rows of boxes as word (with spaces between them
<lang j>[:</lang> placeholder meaning "no verb here" the verb to the right does not get a left argument
<lang j>|."_1</lang> rotate each item (each row of characters) by the indicated amount (rotation would be moving contents left except that the amounts are negative).
<lang j>-@|.</lang> reverse and negate the argument (resulting in the negative sequence _4 _3 _2 _1 0)
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