Partition Rank
Return Dyson’s rank, largest part minus number of parts, for a nonincreasing positive-integer partition; the empty partition has rank zero and input is capped at 1,000 parts.
Description
Dyson’s partition rank is the largest part minus the number of parts.
Dyson’s partition rank is the largest part minus the number of parts. It is a statistic of a supplied partition rather than the count of partitions of its total.1
The tool returns rank. The indexing and output conventions below determine how to read these values.
When to use Partition Rank
Use this for the following task: Return Dyson’s rank, largest part minus number of parts, for a nonincreasing positive-integer partition; the empty partition has rank zero and input is capped at 1,000 parts. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Partition Rank and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter parts according to the field table, then run the tool.
- Read rank. Use the stated units, indexing and interpretation of the numerical result.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| parts | Partition parts in nonincreasing order | list of integer; minimum entries 0; maximum entries 1000; each entry: minimum 1; maximum 9007199254740991 | Required |
Require positive safe-integer parts in nonincreasing order, with at most 1000 parts. Return zero for the empty list and the stated difference otherwise.
rank = largestPart−partCount; empty rank = 0 The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| rank | Dyson partition rank | integer |
{
"input": {
"parts": [
4,
2,
1
]
},
"output": {
"rank": 1
}
}Limitations and assumptions
The input is not automatically sorted. Different partitions of the same total can have different ranks, and no partition count is calculated.
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 Partition Crank when you need its operation: Return the partition crank: largest part if no ones occur, otherwise the number of parts greater than the number of ones minus that number; empty crank is zero. Choose Integer Partition Number for the complementary 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.
Background reference: OEIS A340601.
References
-
OEIS A340601 — OEIS Foundation
Similar or alternative tools
- Partition Crank
Return the partition crank: largest part if no ones occur, otherwise the number of parts greater than the number of ones minus that number; empty crank is zero.
- 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.