Combination Prefix Completion Count
Count exact sorted-combination completions of an explicit prefix, excluding earlier skipped indexes; does not enumerate or assume the prefix is arbitrary subset membership.
Description
Count exact sorted-combination completions of an explicit prefix, excluding earlier skipped indexes; does not enumerate or assume the prefix is arbitrary subset membership.
A sorted combination prefix restricts which later elements can complete it. Counting completions avoids enumerating them.
When to use this tool
Count exact sorted-combination completions of an explicit prefix, excluding earlier skipped indexes; does not enumerate or assume the prefix is arbitrary subset membership.
- Prepare the supplied prefix, size, count fields for this operation.
- Compare the complete example with the representation used by your own input before carrying the result into another step.
How to use the tool and how it works
- Enter prefix, size, count. Use the exact text syntax, units and index conventions explained below.
- Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
- Read result.completions. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Validate increasing prefix indexes in a universe of size at most 128. Require count between prefix length and size, count positions after the last prefix element and return choose(available, count minus prefix length) as exact decimal text.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| prefix | prefix. | List of finite numbers; at most 128 entries |
| size | size. | Safe integer |
| count | count. | Integer from prefix.length to size inclusive |
| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.completions | The completions value under the stated formula or text convention. |
{
"input": {
"prefix": [
1,
3
],
"size": 6,
"count": 3
},
"output": {
"result": {
"completions": "2"
}
}
}Limitations and assumptions
- count is the desired total combination size. An impossible suffix count returns zero; the tool does not complete or reorder the prefix.
- All declared input fields are required and unexpected top-level fields are rejected. Numbers must be finite; lists are limited to 128 entries and ordinary text fields to 8,192 UTF-16 code units, with tighter limits where stated.
- JSON text inputs must parse within bounded nesting and size limits. Numeric outputs outside the safe finite range are rejected; large exact integer results use decimal strings. Indexes and half-open spans are zero-based unless a rank is explicitly described as one-based.
Alternative or Complementary methods
Retain the universe size, subset size and ordering convention with a rank. Use rank/unrank pairs from the same convention and distinguish subset masks from lexicographic indexes.
References
Similar or alternative tools
- Combination Colexicographic Rank
Exact zero-based colexicographic combination ranking on bounded sorted distinct indexes; results are integer text or explicit index witnesses.
- Combination Colexicographic Unrank
Exact zero-based colexicographic combination unranking on bounded sorted distinct indexes; results are integer text or explicit index witnesses.