Why are the catalan numbers giving the unique/correct patterns from all the combinations?

Question

I am reading about catalan numbers and they are considered to represent the number of valid pair of parentesis, mountains etc.
Although the number checks out correct when comparing against specific cases e.g. $n = 2$ or $n = 3$ for pairs of parenthesis, it is not clear to me how it is correct.
I am reading an example in a book using mountains i.e. strokes up and down e.g.

       /\/\
/\/\  /
    \/   

and mentions that if we add an extra up stroke we have $\frac{7!}{4!3!} = 35$ sequences of $4$ upstrokes and $3$ downstrokes which I think means that if we have in total $7$ symbols and we choose $3$ or $4$ pairs (${7 \choose 3} = {7 \choose 4}$) we have $35$ combinations valid and invalid in total e.g. for the example of parenthesis something like $)()($ or $)((($is invalid.
Then the text claims that if we continue these patterns periodically we get only $5$ infinite sequences i.e. only $5$ different mountains with $3$ upstrokes and $3$ downstrokes.
I don't really understand this point.

1. How is it that we are guaranteed that we get only the number of valid configurations?
2. Trying to reproduce the process with a small number e.g. $4 \choose 2$ which gives $6$ possible configurations, and started appending and repeating them I couldn't come with just $2$ unique patterns repeating so I must be doing something wrong

Can someone help understand this?

Update:
For example I was trying to follow the process with a small number i.e. $4 \choose 2$ which gives $6$ which are (map opening bracket to upstroke and closing to downstroke):

()() or /\/\  (1)

         /\
(()) or /  \  (2)


          /\  (3)
)(() or \/



())( or /\    (4)
          \/  



)()( or  \/\/  (5)


))(( or \  /   (6)
         \/
   

According to the text if I append them there should be only $2$ infinite patterns but I don't understand how to reproduce the process.

E.g I can start appending $1,3,3,4,5,2,3,1,3,3...$ to form one sequence but I am not sure what criteria I can use to figure out if I am doing this right

 1    3   3   4  5   2    3

                     /\
/\/\  /\  /\/\      /  \  /\
    \/  \/    \/\/\/    \/

Answer 1

We already have an idea of the nice relationship between expressions of well-formed parentheses and mountains. Let's revisit the relevant text part in The Book of Numbers by J.H. Conway and R.K. Guy.

• [Conway, Guy]: To see that they are the Catalan Numbers, it's perhaps easiest to count the mountains. ... If we add in an extra upstroke, there are $7!/4!3!=35$ sequences of $4$ upstrokes and $3$ downstrokes, but if we continue these patterns periodically ... we get only $5$ different infinite sequences, which break naturally at the dashed edges to reveal the $5$ different mountains with $3$ upstrokes and $3$ downstrokes.

I think the interesting part, which might help to clarify the situation are the words: ... which break naturally at the dashed edges ....

The graphic shows the case $n=2$ with $C_n=2$. The upper two mountain chains are periodic copies of well formed mountains corresponding to the well-formed parentheses. A well-formed mountain

• starts with an upstroke,

• has as many upstrokes as downstrokes and

• each left part of a mountain contains at least as many upstrokes as downstrokes.

This way up- or downstrokes of a mountain are never below its baseline, the segment joining leftmost and rightmost endpoint.

We observe the two valid mountain chains have the nice property, that each mountain is connected with an upstroke and we can draw a dotted line going through the left point of dashed edges. Invalid mountain chains do not have this property as seen in the third part of the graphic.