Partitions into kth Powers

Count unordered partitions of a nonnegative total into positive kth powers with repetition, using exact counts, any positive safe-integer exponent, and a total limit of 2,000.

Description

Partitions into kth powers restrict every positive part to a perfect kth power and permit repetitions.

Partitions into kth powers restrict every positive part to a perfect kth power and permit repetitions. Part order remains irrelevant.^[1]

The tool returns value. The indexing and output conventions below determine how to read these values.

When to use Partitions into kth Powers

Use this for the following task: Count unordered partitions of a nonnegative total into positive kth powers with repetition, using exact counts, any positive safe-integer exponent, and a total limit of 2,000. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Partitions into kth Powers and understand its results

Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
Enter total, power 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
total	Nonnegative total (resource limit 2000)	integer; minimum 0; maximum 2000	Required
power	Positive integer power	integer; minimum 1; maximum 9007199254740991	Required

Generate powers only while they fit the total, using bounded integer multiplication to avoid overflow. Add each allowed part by upward BigInt count updates.

count = coefficient of q^total in Π[b≥1](1−q^(b^power))⁻¹

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
value	Partition count using positive kth-power parts	string (integer/arbitrary)

json

{
  "input": {
    "total": 10,
    "power": 2
  },
  "output": {
    "value": "4"
  }
}

Limitations and assumptions

The total is at most 2000. The positive exponent may be large, but power generation stops at the bound; zero has one empty partition.

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 Overpartition Count for the complementary operation: 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.

Background reference: Digital Library of Mathematical Functions, §26.9.

References

Digital Library of Mathematical Functions, §26.9 — National Institute of Standards and Technology

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.
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.