Description
Check Dyck-word balance and prefix nonnegativity.
Graph and lattice counts depend on the allowed moves, edges and geometric or incidence assumptions. A necessary count identity or local conflict test may hold even when no valid complete object with the requested structure has been supplied.
Check Dyck-word balance and prefix nonnegativity.
When to use this tool
Use Combinatorics: Dyck Word Up Down Balance when checking one supplied path, tree, matching or rook-placement condition.
How to use the tool and how it works
- Enter word. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Check Dyck-word balance and prefix nonnegativity. Accept only U/D characters; accumulate +1/-1, return finalBalance, minimumPrefixBalance and dyck when final zero with no negative prefix.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| word | Word | Required. Text. Implementation limit: 8192 Unicode code points and 16384 UTF-16 code units. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Returned calculated value or nested object under the stated output convention. | Required. Object. |
| result.finalBalance | Final Balance | Calculated result member: number. |
| result.minimumPrefixBalance | Minimum Prefix Balance | Calculated result member: number. |
| result.dyck | Dyck | Calculated result member: boolean. |
{
"input": {
"word": "UUDD"
},
"output": {
"result": {
"finalBalance": 0,
"minimumPrefixBalance": 0,
"dyck": true
}
}
}Limitations and assumptions
- The empty word is valid; other alphabets and constrained Dyck-path counting are outside scope.
- Outputs are nested inside result. Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
Alternative or Complementary methods
Inspect or construct the complete graph/path and apply all relevant structural constraints for a full validity or counting result.
References
Similar or alternative tools
- Dyck Prefix Next Step Feasibility
Return next-step feasibility from supplied Dyck-prefix counts.
- Combinatorics: Lattice Path Endpoint Parity Check
Check square-lattice endpoint reachability for an exact step count.