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Counting, Permutations & Combinations

Catalan Number Calculator

Calculate Catalan numbers for lattice paths, balanced structures, and related counting problems.

Counting & Combinatorics

Calculate a Catalan number

Exact Cₙ
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example

Exact count

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Understand the subject

What do Catalan numbers count?

Catalan numbers appear when a structure must stay balanced or avoid crossing a boundary. Different-looking problems can share this exact rule.

Use it only after matching your problem to a Catalan structure such as balanced parentheses or monotone paths constrained by a diagonal.

The relationship

The counting rule

See the structure

Balance is the extra restriction

Worked example

Check the rule with a small case

The fifth Catalan number is C5 = C(10,5) ÷ 6 = 42.

Interpret with care

Choose the right counting model

A central binomial coefficient is not automatically Catalan; a non-crossing or never-below-boundary rule is essential.

Use the Permutation & Combination Generator when a small case needs to be inspected rather than only counted.

Quick guide

How to use this calculator

  1. Enter a non-negative index n.
  2. Read the exact Catalan number Cₙ.
  3. Apply it only when your problem matches a Catalan-counted structure.

Calculation method

Divide the central binomial coefficient

The nth Catalan number is C(2n,n)/(n+1).

Worked example

Fifth Catalan number

The fifth Catalan number is 42.

C₅ = C(10,5)/6 = 42

Supported inputs

Precision and limits

Discrete domain

Inputs are whole counts. Negative, fractional, grouped, and scientific-notation inputs are rejected.

Exact integers

Results use arbitrary-precision integers and are never rounded. Inputs and output size are bounded to keep the page responsive.

Interpretation

Catalan numbers count many different structures, but only when the structure satisfies the corresponding restrictions.