Math Is Fun Forum

Hello!

I have this problem in combinatorics which I need to solve, but I don't know where to look for information on it. Maybe someone here has seen something similar and can direct me to the right place?

The problem is the following. Suppose you are given n1 copies of the number "1", n2 copies of the number "2" and n3 copies of the number "3". You then arrange them in a sequence of length n1+n2+n3. Then you scan the sequence and write down the transitions that you encounter.

For example, if you write 1123312 the transitions are (11)(12)(23)(33)(31)(12)

Now comes the question. Suppose I want to fix the number of transitions, i.e. I require that the transition (jk) appears a number m(jk) of times. Question is: How many different sequences can I write with the same m(jk), given n1, n2 and n3.

Of course the numbers must satisfy some conditions, such as the sum of all m(jk) equals n1+n2+n3-1 and m(12)<2min(n1,n2).

For a simple example, for n1=n2=n3=1 with m(11)=m(22)=m(33)=m(13)=0 and m(12)=m(23)=1 there are two possibilities, 123 - 321

I would really appreciate some help!
Thanks!