Dyck Path Counter

How it is calculated

The number of Dyck paths of semilength n is the Catalan number C(n) = (2n)! / (n! × (n+1)!), which is the same as C(2n, n) / (n+1). Instead of using large factorials, the tool builds it step by step with C(0) = 1 and C(k) = C(k−1) × 2(2k−1) / (k+1), using exact whole-number arithmetic. A path of semilength n has n up steps and n down steps, so 2n steps in total, which the tool also reports. Only whole numbers from 0 to 30 are accepted: C(30) = 3,814,986,502,092,304 is still exact, but larger values are not computed, and decimals, negatives or an empty field show no result. It also echoes your n.

Common mistakes

• Entering the total number of steps (8) instead of the semilength (4). The tool expects n, the number of up steps, and doubles it for the length.
• Confusing Dyck paths with all up/down paths. Without the rule that the path never goes below the start, there would be C(2n, n) paths, which for n = 4 is 70, not 14.
• Entering a decimal such as 2.5, or a negative n. Only whole numbers from 0 up are valid, and anything else gives no result.
• Entering n above 30 and expecting a very large number. The tool stops at 30 to keep the count exact, so for bigger n you need a big-integer tool.
• Assuming n = 0 gives zero. There is exactly one empty path, so the count for n = 0 is 1 (with 0 steps).