Catalan Path Counting Calculator

Work out catalan path counting instantly with clear inputs, formula shown and shareable results.

About Catalan Path Counting Calculator

On an n × n grid there are C(2n, n) monotonic paths from corner to corner. Exactly Cₙ = C(2n, n)/(n + 1) of them stay weakly below the diagonal, which is the classic Catalan interpretation.

Formulas

Catalan paths

Cₙ = C(2n, n) / (n + 1)

All monotonic paths

C(2n, n)

Worked example

With the default values below, Catalan Path Counting Calculator returns Paths of the chosen kind of 132. Change any input above and every figure here is recalculated in your browser.

Inputs used

Grid size naccepts 0–30: 6
Countone of Monotonic paths staying below the diagonal, All monotonic lattice paths: Monotonic paths staying below the diagonal

Results

Paths of the chosen kind: 132
Catalan number Cₙ: 132
All monotonic paths C(2n, n): 924
Paths that cross the diagonal: 792

Step by step

1Values used
2Catalan paths
3All monotonic paths
4Paths of the chosen kind= 132
5Catalan number Cₙ= 132
6All monotonic paths C(2n, n)= 924
7Paths that cross the diagonal= 792

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For a 6 × 6 grid?

924 monotonic paths in total, of which 132 stay below the diagonal.

What else does Cₙ count?

Balanced bracket strings of length 2n, binary trees with n nodes and triangulations of an (n + 2)-gon — all the same number.