Description
Uniformly sample a fixed-pair Dyck word by exact valid-completion count unranking; no rejection of unbalanced random strings, bounded to 64 pairs.
A random construction is defined by its sampling support: positions, permutations, complete tilings, proposal sequences or another explicit set. Uniform choices at one stage do not necessarily make final visible outcomes uniform. A replay seed gives a deterministic sequence; equal values can hide different sampled positions. These tools are for reproducible synthetic examples rather than cryptographic entropy.
Uniformly sample a fixed-pair Dyck word by exact valid-completion count unranking; no rejection of unbalanced random strings, bounded to 64 pairs.
When to use this tool
Use this tool to uniformly sample a fixed-pair Dyck word by exact valid-completion count unranking; no rejection of unbalanced random strings, bounded to 64 pairs.
How to use the tool and how it works
- Enter pairs, seed using the definitions below.
- Run the operation. Resolve input-format errors before interpreting an output.
- Read result.seed, result.word, result.support, using the stated boundary, ordering, sign and absence conventions.
Count valid balanced-parenthesis completions by open/closed counts, draw an exact uniform rank, then unrank by testing the opening branch’s completion count before the closing branch. Return the Dyck word and exact support.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| pairs | Number of balanced parenthesis pairs, integer 1–64. | Finite safe integer satisfying the stated bounds. |
| seed | Nonblank replay seed, at most 256 UTF-16 code units; an injected source overrides seeded draws. | Text, at most 8,192 UTF-16 code units; narrower grammar/range rules also apply. |
| Output field | Interpretation |
|---|---|
| result | Object containing the fields below. |
| result.seed | Original supplied seed string; injected-source replay additionally requires that source’s state. |
| result.word | Complete balanced-parenthesis Dyck word. |
| result.support | Exact finite support count as decimal text. |
{
"input": {
"pairs": 3,
"seed": "batch56"
},
"output": {
"result": {
"seed": "batch56",
"word": "()(())",
"support": "5"
}
}
}Limitations and assumptions
- pairs is 1–64. Every prefix remains balanced; arbitrary equal-count parenthesis strings outside the Dyck support are excluded.
- A nonblank seed of at most 256 UTF-16 code units is required. With ordinary use the seed initializes one deterministic advancing source; an application-injected random source takes precedence and the seed alone then cannot replay the result. Uniformity statements assume independent uniform source draws in [0,1). Source calls are bounded and degenerate injected sources can fail a rejection budget. This is not cryptographically secure.
- Every declared outer field is required and unknown outer fields are rejected. Strings and JSON text have size limits, lists are bounded to 128 entries, and JSON nesting/node counts are bounded. Tighter operation-specific bounds stated above take precedence.
- Ordinary numeric outputs must be finite and within safe-integer magnitude. Exact large lengths/counts use decimal strings. Original record indexes and code-unit positions are zero-based; span ends are exclusive unless stated otherwise.
Alternative or Complementary methods
Choose a generator whose support matches the intended experiment and retain source-position provenance. Use a separate statistically justified process model or cryptographic source where those properties matter.
References
Similar or alternative tools
- Random Complementary Bitmap Pair
Sample one fair independent-cell binary bitmap and return its exact bitwise complement; pair assignments uniform over first masks, not independent paired images.
- Random Fixed-Endpoint Unit Walk
Uniformly sample all unit-step walks with specified start/end and step count by choosing exactly the required up-step subset; impossible endpoint/parity requests reject.