Show that the Catalan numbers count the following

Asked May 04 '20 at 04:23

Active May 04 '20 at 04:23

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Show that the Catalan numbers count the following:

1) Sequences $x_1x_2...x_n$ of positive integers such that $x_1\leq x_2 \leq ... \leq x_n$ and for each $i, x_i \leq i.$ Ex) When n = 3, the sequences counted are 111,112,113,122,123

2)Ways to join 2n points on a circle with n non-intersecting chords.

I believe for #1 I need to show there is a bijection through the lattice path and for #2 I need to show a bijection through the balanced set of paraenthesis. However, I do not understand how to do so. I have not come across a bijection proof of this sorts. Any help would be appreciated

asked May 04 '20 at 04:23

For (1), think of the region underneath a lattice path as a "bar graph" consisting of bars of width 1 and heights $x_i$. For (2), wrap a balanced parentheses string around a circle and draw chords connecting the matched pairs of parentheses. – Karl May 04 '20 at 04:59
Suppose you have $a_i$ copies of each integer $i=1,\dots,n$, so that $a_1+\dots+a_n=n$. Construct a Dyck path from $(0,0)$ to $(n,n)$ with unit steps north and east, whose peaks are at $\displaystyle\left(\sum_{j=1}^{i}{a_j},\ i-1\right)$ for those $i\in[1,n]$ for which $a_i>0$. – Alexander Burstein May 04 '20 at 06:14
For a hint towards $(1)$, see counting monotonically increasing sequence that for each object a(i) < i. – Mike Earnest May 04 '20 at 17:33
i'm still stuck on #1, if anyone can help – May 04 '20 at 19:34

Show that the Catalan numbers count the following

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