Maths mini-project

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Evgeny E.Joseph S.Abu Bakar Siddique K.David C. have already sent a proposal.
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Experience Level: Expert
This is a nice mini-project for a mathematician - one who knows her/his combinations and permutations and some number theory. Please don\'t apply if you are not familiar with this branch of maths.

This is probably only a few hours work for a competent person in this field.

But if successful, there may well be more work of a similar nature. Also, you might actually enjoy doing the task!

Imagine a w x h matrix, every cell of which can be filled with a single character from a character set of base b, that is n characters in all, where n=wh.

Typically, 6 <= b <= 20 and 3 <= w,h <= 8 , so 9 <= n <= 64

I have worked out for myself that there are a maximum possible
T = 10 x 9 x 8 x 7 x 6 x 5 = 151,200 sets of 6 numbers between 0 and 9 that can populate a 3 x 2 matrix, with no digit repeated within a set.

At this stage, I am not interested in the position of the characters in the matrix.

So (1,2,3,4,5,6) is the same set as (6,5,4,3,2,1).

However, where n > b, things get more complicated, as we need to repeat some characters in order to fill the matrix.

The first thing I need to know is what is the formula for T where we allow repetition of characters. Don\'t forget that different characters could be repeated (r1, r2, r3, . . . . . ,rb) times.

Clearly, all values of r must fall into limited ranges, depending on whether n < b or n >b

So your first task is to find formulae to express the relationships between
T, n, b and r1 . . . . rb, for all values within the stated typical ranges (including where n<b and there is repetition).

The second task is to consider how position within the matrix changes things.

Clearly, T will be a larger number if we say that (1,2,3), (1,3,2), (2,1,3) etc are
now different sets. But what is the maximum possible number of arrangements, T, expressed again as a relationship between n,b and r1, r2, etc.?

I imagine, in doing this task, you will have questions and observations. Feel free to contact me by email at any time to discuss these.

In your bid, please refer to any work you have done of a similar nature and your qualifications for doing this work.

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