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.
Description
The partition crank uses the largest part when no ones occur, and a count-based rule when ones are present.
The partition crank uses the largest part when no ones occur, and a count-based rule when ones are present. It captures information different from Dyson’s rank.1
The tool returns crank. The indexing and output conventions below determine how to read these values.
When to use Partition Crank
Use this for the following task: 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. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Partition Crank 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 crank. 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 (resource limit 1000) | list of integer; minimum entries 0; maximum entries 1000; each entry: minimum 1; maximum 9007199254740991 | Required |
Validate a nonincreasing list of positive safe-integer parts with at most 1000 entries. Count ones, apply the matching branch and return zero for an empty partition.
if ones=0: crank=largestPart; otherwise crank=count(parts>ones)−ones The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| crank | Partition crank | integer |
{
"input": {
"parts": [
3,
2,
1,
1
]
},
"output": {
"crank": -1
}
}Limitations and assumptions
The tool computes this direct partition statistic, without sorting, counting partitions or applying generating-function exceptional conventions for small totals.
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 Rank when you need its operation: 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. 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 A064410.
References
-
OEIS A064410 — OEIS Foundation
Similar or alternative tools
- 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.
- 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.