Triangulations of an n-gon

Task number: 3877

How many distinct triangulations of a regular \(n\)-gon with labeled vertices exist?

  • Solution

    Denote the vertices cyclically \(A_1,A_2,\ldots,A_n\) and consider in the triangulation the triangle containing the edge \(A_1A_2\). Let its third vertex be \(A_k\), \(k\in\{3,\ldots,n\} \). The number of possible triangulations containing the triangle \(A_1,A_2,A_k\) is then equal to \(t_{k-1}t_{n-(k-2)}\) Here \(t_{k-1}\) is the number of triangulations of the \((k-1)\)-gon \(A_2,A_3,\ldots,A_k\) and \(t_{n-(k-2)}\) is the number of triangulations of the \(n-(k-2)\)-gon \(A_1,A_k,\ldots,A_n\).

    For the case when the triangle \(A_1,A_2,A_k\) has two edges incident with the boundary, we set \(t_2=1\). We obtain the recurrence:

    \(\displaystyle t_n=\sum_{k=3}^n t_{k-1}t_{n-(k-2)} = \sum_{k=2}^{n-1} t_k t_{n+1-k}. \)

    For better clarity, we rewrite without the sum

    \( t_n=t_2t_{n-1}+t_3t_{n-2}+\ldots+t_{n-2}t_3+t_{n-1}t_2\)



    After the substitution \(b_n=t_{n+2}\) we obtain the recurrence:


    It can be seen that we get the same recurrence as for the Catalan numbers \(b_n\) including the same initial conditions, i.e. \(b_0=t_2=1\).

  • Answer

    The given \(n\)-gon can be triangulated in \(\frac{1}{n-1}\binom{2n-4}{n-2}\) distinct ways.

Difficulty level: Moderate task
Solution require uncommon idea
Complex task
Cs translation
Send comment on task by email