Discussion:
Which formula to use?
(too old to reply)
Patrick D. Rockwell
2015-01-27 01:13:26 UTC
Permalink
Raw Message
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.

MMWWW
MWMWW

Now let's try the same thing for with 4 men
and 6 women. I've counted 23 ways.

MMMMWWWWWW
MMMWMWWWWW
MMMWWMWWWW
MMMWWWMWWW
MMMWWWWMWW
MMWMMWWWWW
MMWMWMWWWW
MMWMWWMWWW
MMWMWWWMWW
MMWWMMWWWW
MMWWMWMWWW
MMWWMWWMWW
MMWWWMMWWW
MMWWWMWMWW
MWMMMWWWWW
MWMMWMWWWW
MWMMWWMWWW
MWMMWWWMWW
MWMWMMWWWW
MWMWMWMWWW
MWMWMWWMWW
MWMWWMMWWW
MWMWWMWMWW

Is there a way to calculate these numbers?
I've tried the fuss catalan formula and the
ballot theorem without success. Have I
miscounted?
William Elliot
2015-01-27 02:49:42 UTC
Permalink
Raw Message
I've crossed posted your problem to sci.math.
Post by Patrick D. Rockwell
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.
MMWWW
MWMWW
Now let's try the same thing for with 4 men
and 6 women. I've counted 23 ways.
MMMMWWWWWW
MMMWMWWWWW
MMMWWMWWWW
MMMWWWMWWW
MMMWWWWMWW
MMWMMWWWWW
MMWMWMWWWW
MMWMWWMWWW
MMWMWWWMWW
MMWWMMWWWW
MMWWMWMWWW
MMWWMWWMWW
MMWWWMMWWW
MMWWWMWMWW
MWMMMWWWWW
MWMMWMWWWW
MWMMWWMWWW
MWMMWWWMWW
MWMWMMWWWW
MWMWMWMWWW
MWMWMWWMWW
MWMWWMMWWW
MWMWWMWMWW
Is there a way to calculate these numbers?
I've tried the fuss catalan formula and the
ballot theorem without success. Have I
miscounted?
quasi
2015-01-27 08:59:12 UTC
Permalink
Raw Message
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.
MMWWW
MWMWW
Now let's try the same thing for with 4 men and 6 women. I've
counted 23 ways.
MMMMWWWWWW
MMMWMWWWWW
MMMWWMWWWW
MMMWWWMWWW
MMMWWWWMWW
MMWMMWWWWW
MMWMWMWWWW
MMWMWWMWWW
MMWMWWWMWW
MMWWMMWWWW
MMWWMWMWWW
MMWWMWWMWW
MMWWWMMWWW
MMWWWMWMWW
MWMMMWWWWW
MWMMWMWWWW
MWMMWWMWWW
MWMMWWWMWW
MWMWMMWWWW
MWMWMWMWWW
MWMWMWWMWW
MWMWWMMWWW
MWMWWMWMWW
Is there a way to calculate these numbers? I've tried the
fuss catalan formula and the ballot theorem without success.
Have I miscounted?
No.

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:

1, 2
2, 23
3, 377
4, 7229
5, 151491
6, 3361598
7, 77635093
8, 1846620581
9, 44930294909
10, 1113015378438
11, 27976770344941
12, 711771461238122
13, 18293652115906958
14, 474274581883631615
15, 12388371266483017545
16, 325714829431573496525
17, 8613086428709348334675
18, 228925936056388155632081
19, 6112355595348903438948155
20, 163869604996054172670563730

quasi
Don Reble
2015-01-27 09:42:00 UTC
Permalink
Raw Message
http://oeis.org/A060941
--
Don Reble ***@nk.ca
Charlie Roberts
2015-01-28 03:39:22 UTC
Permalink
Raw Message
Post by Don Reble
http://oeis.org/A060941
Whoa! Amazing!!

Did you work in this?

cr

John Jones
2015-01-27 10:20:24 UTC
Permalink
Raw Message
Post by quasi
Post by William Elliot
I've crossed posted your problem to sci.math.
Starting with 2n men and 3n women, let f(n) be the number of
legal ordered selections. Then for 1 <= n <= 20, the pairs
snip

brilliant stuff.
fyi Don Keble found
<http://oeis.org/A060941>
posted on the allbut defunct alt.math.recreational.
(I dont know how to forward posts on this ng reader)
cheers
JJ
Loading...