Partition Transform Calculator
Compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation.
Description
A formal Euler-product partition transform maps an exponent sequence to polynomial coefficients.
A formal Euler-product partition transform maps an exponent sequence to polynomial coefficients. Signed exponents are allowed and differ from simply evaluating a numeric product at q.1
Compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation.
When to use Partition Transform Calculator
Use this tool to compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Partition Transform Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter integerEulerProductExponents using the field table and calculation rules, then run the tool.
- Read partitionProductCoefficients. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| integerEulerProductExponents | Integer euler product exponents | list of number | Required |
Supply 1–64 signed integer exponents a1 through aN, each with absolute value ≤one million. Compute exact BigInt coefficients through order N and return decimal strings, including initial coefficient 1. An exact divisibility failure is rejected.
b0=1; c_k=Σ[d divides k]d a_d; b_n=(1/n)Σ[k=1..n]c_k b_(n−k) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| partitionProductCoefficients | Partition product coefficients | list of string |
{
"input": {
"integerEulerProductExponents": [
1,
1,
1,
1
]
},
"output": {
"partitionProductCoefficients": [
"1",
"1",
"2",
"3",
"5"
]
}
}Limitations and assumptions
The output is a finite formal-series prefix. It is not a general partition-count lookup or an infinite numeric product.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Partition Generating Function Approximation: Evaluate truncated real generating product ∏_(k=1..N)(1−q^k)^−1 with stable log accumulation, not exact partition coefficient counting or the exponent-sequence transform. For a complementary task, use Bell Transform Calculator: Compute complete exponential Bell-polynomial transform with b0=1 and b_n=Σ C(n−1,k−1)a_k b_(n−k); caller sequence differs materially from a single Bell number.
References
-
Partition Transform Calculator — tool.io
Similar or alternative tools
- Partition Generating Function Approximation
Evaluate truncated real generating product ∏_(k=1..N)(1−q^k)^−1 with stable log accumulation, not exact partition coefficient counting or the exponent-sequence transform.
- Bell Transform Calculator
Compute complete exponential Bell-polynomial transform with b0=1 and b_n=Σ C(n−1,k−1)a_k b_(n−k); caller sequence differs materially from a single Bell number.