ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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The five Dyck paths of semilength $3$ , with their height functions

Example

The five Dyck paths of semilength $3$ are:

step word	height sequence
`UUUDDD`	$0, 1, 2, 3, 2, 1, 0$
`UUDUDD`	$0, 1, 2, 1, 2, 1, 0$
`UUDDUD`	$0, 1, 2, 1, 0, 1, 0$
`UDUUDD`	$0, 1, 0, 1, 2, 1, 0$
`UDUDUD`	$0, 1, 0, 1, 0, 1, 0$

Facts & Assumptions

Given: the Dyck paths of semilength $3$ .

[L1]

$C_{3}$ is the number of Dyck paths of semilength $3$ (The Catalan number $C_{n} : = ∣ D_{n} ∣$ ).

[L2]

The closed formula gives $(3 + 1) C_{3} = (\binom{6}{3})$ ( $(n + 1) C_{n} = (\binom{2 n}{n})$ ).

Verification

technique · direct

1.1

Every word in the table has three up steps, three down steps, starts at height $0$ , ends at height $0$ , and never drops below height $0$ , so every row is a Dyck path of semilength $3$ .

given

2.1

Any Dyck path of semilength $3$ must begin with U; listing the five possible continuations that keep the height nonnegative gives exactly the five rows of the table and no others.

step 1.1

3.1

The table therefore has all the Dyck paths of semilength $3$ , so [L1] gives $C_{3} = 5$ . This matches [L2], since $(\binom{6}{3}) = 20$ and therefore $4 C_{3} = 20$ .

L1L2step 2.1∎

Remarks

The five words are the first nontrivial Catalan family large enough for the reflection, cycle-lemma and triangulation examples to display all members explicitly.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Postnikov (notes by A. Lin), MIT 18.212 Algebraic Combinatorics, lecture of February 6, 2019 (standard reference, not scraped)

The five Dyck paths of semilength 33, with their height functions