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.

Description

A plane partition is a two-dimensional arrangement of positive parts that decreases across rows and columns.

A plane partition is a two-dimensional arrangement of positive parts that decreases across rows and columns. Counting these differs from ordinary one-dimensional integer partitions.^[1]

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

When to use Plane Partition Count

Use this for the following task: Count plane partitions of n exactly from MacMahon’s generating product ∏(1−q^k)^(−k), including the empty partition; n is capped at 200. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Plane Partition 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 200)	integer; minimum 0; maximum 200	Required

Build a BigInt coefficient array through n, adding unlimited parts of size k in k distinguishable factor passes. Read the coefficient of total n.

generating function: Σpp(n)q^n = Π[k≥1](1−q^k)^(−k)

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

Output field	Meaning	Value type
value	Plane partition count	string (integer/arbitrary)

json

{
  "input": {
    "n": 4
  },
  "output": {
    "value": "13"
  }
}

Limitations and assumptions

The total is capped at 200. Zero has one empty plane partition, and the tool returns a count without producing arrays or imposing a box shape.

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, §26.12.

References

Digital Library of Mathematical Functions, §26.12 — 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.