# What kind of recursion or series is this?

I am developing a language learning program, and would like to know the mathematical expression and description for what I have (perhaps incorrectly) calling a recursive series method of learning the order of a foreign alphabet.

In sum, the learner would write out the new alphabet (I'll use English as an example) in order, one line at a time:

Example:

a
ab
abc
abcd
{...}
abcdefghijklmnopqrstuvwx
abcdefghijklmnopqrstuvwxy
abcdefghijklmnopqrstuvwxyz


In initial tests, this has shown to be a highly time-efficient way of getting illiterate students to accurately recall the name, sound, and order of an alphabet.

As I consider further research and even publication, I would like to know what to call this pattern, in mathematical language.

If it helps, the pattern is basically

From 1 to 10

1
1,2
1,2,3
1,2,3,4
1,2,3,4,5
1,2,3,4,5,6
1,2,3,4,5,6,7
1,2,3,4,5,6,7,8
1,2,3,4,5,6,7,8,9
1,2,3,4,5,6,7,8,9,10


Notice that each "column" of numbers is written multiple times (except for the final number, of course) and the ascending order of the series is repetitive enough that a reader/student will learn the order of whatever ordered set you throw at him in a short amount of time.

Basically: is there a way of using algorithmic notation to say: given a full set (in this case, the alphabet), iterate through each element along with its preceding elements, until you have displayed the whole set?

I keep thinking of it being represented as something like f(n) = f(n-1) + f(n-2)... but I don't want to just guess at it.

I appreciate your help in describing these patterns algorithmically, and in terms of what kind of recursion is going on there.

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The order is a restriction of the lexicographic (dictionary) order on words. –  André Nicolas Mar 27 '12 at 15:25
That's the sort order, yet, but I'm asking for a mathematical algorithm and descriptor of this pattern (see updated ending) –  Yaaqov Mar 27 '12 at 15:41
You are simply enumerating the initial segments of your ordering. –  Bill Dubuque Mar 27 '12 at 16:26
Incrementally learning the alphabet? –  ShreevatsaR Mar 27 '12 at 17:27
@BillDubuque - I suppose that's basically it, but is there a way of using algorithmic notation to say: given a full set (in this case, the alphabet), iterate through each element along with its preceding elements, until you have displayed the whole set? –  Yaaqov Mar 27 '12 at 18:12

Consider the following.

$$\text{Let } f: \{n: n \leq 26 \text{ and } n \in \mathbb{Z^+} \} \rightarrow \{\text{A, B, C, D, ..., Z}\}, n_i \mapsto x_i.$$

Here, $i \in \text{ dom }\ f$, and $n_i \text{ and } x_i$ denote the element at the ith index in dom $f$ and range $f$, respectively.

Then, let $\{f(n)\}$ denote the set $\{f(1),\ f(2),\ f(3),\ ...,\ f(n)\}$. What you're asking for is accomplished by simply computing $\{\{f(1)\},\ \{f(2)\},\ \{f(3)\},\ ...,\ \{f(26)\}\}$.

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That's it! And what terminology would you use to describe this algorithm? Was I off by thinking of it as something recursive? –  Yaaqov Mar 27 '12 at 19:00
I'm not so sure about the terminology; it's just something I came up with using set theory. As far as recursion goes, I don't think you can call this recursive since no computation at a particular index depends on any previous index. –  ThisIsNotAnId Mar 27 '12 at 19:44
That works for me - thanks for the clarification. –  Yaaqov Mar 27 '12 at 23:54
@Yaaqov Sure, my pleasure. –  ThisIsNotAnId Mar 28 '12 at 3:12