Overpartition Count
Count partitions of n in which the first occurrence of each distinct part may be overlined, returning an exact integer and limiting n to 1,000.
Description
An overpartition allows the first occurrence of each distinct part size to be overlined.
An overpartition allows the first occurrence of each distinct part size to be overlined. For a size that occurs, marked and unmarked choices produce separate objects.1
The tool returns value. The indexing and output conventions below determine how to read these values.
When to use Overpartition Count
Use this for the following task: Count partitions of n in which the first occurrence of each distinct part may be overlined, returning an exact integer and limiting n to 1,000. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Overpartition Count and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter n according to the field table, then run the tool.
- Read value. Preserve exact decimal strings and the stated indexing or coefficient order.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| n | Nonnegative total (resource limit 1000) | integer; minimum 0; maximum 1000 | Required |
For each part size, first perform ascending unlimited-part updates and then descending optional-overline updates using BigInt counts.
generating function = Π[k≥1](1+q^k)/(1−q^k) The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| value | Overpartition count | string (integer/arbitrary) |
{
"input": {
"n": 5
},
"output": {
"value": "24"
}
}Limitations and assumptions
The total is at most 1000. Zero has one empty overpartition; the result is a count and does not render overlined parts.
Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.
Alternative or Complementary methods
Choose Integer Partition Number when you need its operation: Count unordered partitions of n into positive integer parts exactly, including the empty partition of zero; n is capped at 1,000 for bounded computation. Choose Plane Partition Count for the complementary operation: Count plane partitions of n exactly from MacMahon’s generating product ∏(1−q^k)^(−k), including the empty partition; n is capped at 200.
Background reference: OEIS A015128.
References
-
OEIS A015128 — OEIS Foundation
Similar or alternative tools
- Integer Partition Number
Count unordered partitions of n into positive integer parts exactly, including the empty partition of zero; n is capped at 1,000 for bounded computation.
- Plane Partition Count
Count plane partitions of n exactly from MacMahon’s generating product ∏(1−q^k)^(−k), including the empty partition; n is capped at 200.