Description
Uniformly sample complete domino tilings of a 2xN board via exact Fibonacci completion counts and rank unranking, reporting vertical singles or paired horizontal domino placements.
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 complete domino tilings of a 2xN board via exact Fibonacci completion counts and rank unranking, reporting vertical singles or paired horizontal domino placements.
When to use this tool
Use this tool to uniformly sample complete domino tilings of a 2xN board via exact Fibonacci completion counts and rank unranking, reporting vertical singles or paired horizontal domino placements.
How to use the tool and how it works
- Enter width, seed using the definitions below.
- Run the operation. Resolve input-format errors before interpreting an output.
- Read result.seed, result.tiles, result.support, using the stated boundary, ordering, sign and absence conventions.
For width 1–64, count 2×N domino tilings by ways[0]=ways[1]=1 and ways[n]=ways[n−1]+ways[n−2]. Draw an exact rank in the complete support and unrank by choosing a vertical column or paired horizontal dominoes according to completion counts.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| width | Integer width, 1–64, in the positions/cells described above. | 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.tiles | Left-to-right vertical-column or horizontal-pair domino records with starting column. |
| result.support | Exact finite support count as decimal text. |
{
"input": {
"width": 3,
"seed": "batch56"
},
"output": {
"result": {
"seed": "batch56",
"tiles": [
{
"kind": "vertical",
"column": 0
},
{
"kind": "horizontal-pair",
"column": 1
}
],
"support": "3"
}
}
}Limitations and assumptions
- The sample space is complete labeled-board tilings, with each horizontal pair consuming two columns. support is exact decimal text, not a probability.
- 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 Two-Bank Bijection
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