Eulerian Number
Count permutations of n labeled elements with exactly k ascents, using A(0,0)=1 and zero outside the counting range; indices are capped at 300.
Description
Eulerian numbers count permutations by their number of ascents.
Eulerian numbers count permutations by their number of ascents. An ascent is a position where the next element is larger, giving at most n−1 ascents for a nonempty permutation.1
The tool returns value. The indexing and output conventions below determine how to read these values.
When to use Eulerian Number
Use this for the following task: Count permutations of n labeled elements with exactly k ascents, using A(0,0)=1 and zero outside the counting range; indices are capped at 300. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Eulerian Number and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter n, k 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 | Permutation size (resource limit 300) | integer; minimum 0; maximum 300 | Required |
| k | Ascent count (resource limit 300) | integer; minimum 0; maximum 300 | Required |
Handle the empty case as A(0,0)=1 and return zero for impossible ascent counts. Use descending one-row BigInt updates for the recurrence.
A(n,k)=(n−k)A(n−1,k−1)+(k+1)A(n−1,k) The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| value | Eulerian number | string (integer/arbitrary) |
{
"input": {
"n": 4,
"k": 1
},
"output": {
"value": "11"
}
}Limitations and assumptions
Both indexes are at most 300. The count concerns ascents, not cycles or set partitions, and no permutations are enumerated.
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 Signed Stirling Number of the First Kind when you need its operation: Return s(n,k), the cycle-count Stirling number multiplied by (−1)^(n−k), exactly as a decimal integer, with s(0,0)=1 and indices capped at 300. Choose Involution Number for the complementary operation: Count permutations of n labeled elements consisting only of fixed points and transpositions exactly, including the empty involution; n is capped at 2,000.
Background reference: Digital Library of Mathematical Functions, §26.14.
References
-
Digital Library of Mathematical Functions, §26.14 — National Institute of Standards and Technology
Similar or alternative tools
- Signed Stirling Number of the First Kind
Return s(n,k), the cycle-count Stirling number multiplied by (−1)^(n−k), exactly as a decimal integer, with s(0,0)=1 and indices capped at 300.
- Involution Number
Count permutations of n labeled elements consisting only of fixed points and transpositions exactly, including the empty involution; n is capped at 2,000.