[MUSIC] Okay, Dyck paths, D-Y-C-K. These are something like graphs for a sequence. Okay, so we consider a graph Where the x-axis corresponds to People in the line. And the y-axis corresponds to the number of $5 bills in the cash box after the person had obtained their tickets. Okay, so, suppose we are dealing with the first situation. So this is situation 1, and it corresponds to the following path. So at the first moment the cash box is empty. Then, The first guy comes and gives his fiver. So these are 1, 2, and 3. So then, same thing happens with the second one. Then, the third person also pays $5 and gets their ticket. Okay, and what happens next? The fourth person has $10 and after he or she obtains their ticket, there are only two fivers left in the cash box. So the value corresponding to 4 is 2. And then, on the fifth step, the number of fivers in the cash box decreases to 1. And in the sixth moment, it is 0. So let us draw more Dyck paths for the remaining situations, so say for The second sequence. The graph will be as follows, 5, 5. 5 corresponds to one step up, and 10 corresponds to the step going down, 5, 5, 10, 10, 5, 10. So in the third case we see 5, 10, 5, 5. 10, 10. So this is 3. And this is 4. In this case, we have 5, 5, 10, 5, 10, 10. And in the fifth case we have I have a chainsaw, 5, 10, 5, 10, 5, 10. So, We see that Dyck paths are formed by, By line segments. Going up or down. Going by (1,1) or (1, -1) and they're always situated above the x-axis. So the real question is that we need to compute the number of such paths which are formed by segments (1,1) or (1,-1) and situated above the x-axis. And we will now show that they satisfy the same recurrent relation as the number of triangulations. [SOUND]