1. Suppose you begin on a first rung of a ladder. A ladder path is a...

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1. 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 C, be the set of ladder paths with 2n steps, and let C, be the set of Catalan paths from (0,0) to (n, n). Construct a bijection f CnCn. (You don't have to prove that it is a bijection.) 1. 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 C, be the set of ladder paths with 2n steps, and let C, be the set of Catalan paths from (0,0) to (n, n). Construct a bijection f CnCn. (You don't have to prove that it is a bijection.)