Discrete Mathematics

[beachmathguy42069]

Suppose you begin on a first rung of a ladder. A ladder path is a sequence of steps on the ladder where each step is either up one rung or down one rung, you always remain on the ladder, and you end back on the first rung. Assume the ladder is tall enough so that you can never step off the top. For example, there is only one ladder path with two steps, given by taking one step up and then one step down.
(a) Make a list of all the ladder paths with n steps, where n ∈ {3, 4, 5, 6}.
(b) Let n ≥ 1. Write down a correspondence that relates the ladder paths with 2n steps to Catalan paths from (0, 0) to (n, n). More precisely, let Ln be the set of ladder paths with 2n steps, and let Cn be the set of Catalan paths from (0, 0) to (n, n). Construct a bijection f : Ln → Cn. (You don’t have to prove that it is a bijection.)

Anyone know how to do this? Tips to get started?

 

[Deleted member 4993]

Suppose you begin on a first rung of a ladder. A ladder path is a sequence of steps on the ladder where each step is either up one rung or down one rung, you always remain on the ladder, and you end back on the first rung. Assume the ladder is tall enough so that you can never step off the top. For example, there is only one ladder path with two steps, given by taking one step up and then one step down.
(a) Make a list of all the ladder paths with n steps, where n ∈ {3, 4, 5, 6}.
(b) Let n ≥ 1. Write down a correspondence that relates the ladder paths with 2n steps to Catalan paths from (0, 0) to (n, n). More precisely, let Ln be the set of ladder paths with 2n steps, and let Cn be the set of Catalan paths from (0, 0) to (n, n). Construct a bijection f : Ln → Cn. (You don’t have to prove that it is a bijection.)

Anyone know how to do this? Tips to get started?

First step to solution was:

(a) Make a list of all the ladder paths with n steps, where n ∈ {3, 4, 5, 6}.​

Have you done that?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[Dr.Peterson]

Suppose you begin on a first rung of a ladder. A ladder path is a sequence of steps on the ladder where each step is either up one rung or down one rung, you always remain on the ladder, and you end back on the first rung. Assume the ladder is tall enough so that you can never step off the top. For example, there is only one ladder path with two steps, given by taking one step up and then one step down.
(a) Make a list of all the ladder paths with n steps, where n ∈ {3, 4, 5, 6}.
(b) Let n ≥ 1. Write down a correspondence that relates the ladder paths with 2n steps to Catalan paths from (0, 0) to (n, n). More precisely, let Ln be the set of ladder paths with 2n steps, and let Cn be the set of Catalan paths from (0, 0) to (n, n). Construct a bijection f : Ln → Cn. (You don’t have to prove that it is a bijection.)

Anyone know how to do this? Tips to get started?

In addition to showing your answer to part (a), or telling us where you are stuck in doing so, please show us the definition of "Catalan path" as you were taught it, as I suspect there are different ways to define it.

The more you tell us about your own thinking, the more efficiently we can help.

 

[Steven G]

Do you have to stop once you get to the first rung or can you go up again and eventually end at the first rung?
Based on your answer where you have two rungs it seems that the answer is you end the 1st time you reach the 1st rung.
Now do part a.

 

[Dr.Peterson]

Do you have to stop once you get to the first rung or can you go up again and eventually end at the first rung?
Based on your answer where you have two rungs it seems that the answer is you end the 1st time you reach the 1st rung.
Now do part a.

That's not an answer; it's an example (part of the problem); and it implies nothing of the sort. You end when you have taken the required total number of steps:

Suppose you begin on a first rung of a ladder. A ladder path is a sequence of steps on the ladder where each step is either up one rung or down one rung, you always remain on the ladder, and you end back on the first rung. Assume the ladder is tall enough so that you can never step off the top. For example, there is only one ladder path with two steps, given by taking one step up and then one step down.
(a) Make a list of all the ladder paths with n steps, where n ∈ {3, 4, 5, 6}.
(b) Let n ≥ 1. Write down a correspondence that relates the ladder paths with 2n steps to Catalan paths from (0, 0) to (n, n). More precisely, let Ln be the set of ladder paths with 2n steps, and let Cn be the set of Catalan paths from (0, 0) to (n, n). Construct a bijection f : Ln → Cn. (You don’t have to prove that it is a bijection.)

Examples with 3 and 4 steps would be more helpful in clarifying the rules.

But ultimately, your question will probably be answered by the definition of Catalan paths.