Post by William Elliot
I've crossed posted your problem to sci.math.
Let's say that there are 2 men and 3 women randomly entering
a room one at time . You want to make sure that the number
of women in the room is no more than 1.5 times the number of
men in the room. I've counted 2 ways that could happen. If
you read from left to right, that's the order in which they
enter the room.
Now let's try the same thing for with 4 men and 6 women. I've
counted 23 ways.
Is there a way to calculate these numbers? I've tried the
fuss catalan formula and the ballot theorem without success.
Have I miscounted?
It's a Catalan-like recursion, but harder (I think it needs
6 interdependent sequences).
It's not the Catalan formula itself that's needed, but rather
the method by which the Catalan formula was derived, modified
for the constraint 3:2 instead of 1:1.
I don't have time right now to work out a formula, but here's
some data ...
Starting with 2n men and 3n women, let f(n) be the number of
legal ordered selections. Then for 1 <= n <= 20, the pairs
n,f(n) are shown below: