permutations

std.elementwise.permutations · Level L3

The number of ordered selections of k from n distinct objects, P(n, k) = n!/(n − k)!, elementwise, as the falling product n(n − 1)…(n − k + 1): a While over falling_step. P(n, 0) = 1, and P(n, k) = 0 for k > n, since a factor n − n is reached.

P(n, k) = n(n − 1)⋯(n − k + 1)

Signature

permutations(n: i64[s], k: i64[s]) → i64[s]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference math.perm(n, k) in exact rational arithmetic, on all 40 test cases.
All 186 int64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the int64 nearest the exact value): 100%
bit-equal to the NumPy formula in int64: 100%
largest error, in units in the last place: 0

Note

Exact while P(n, k) < 2⁵³: every n ≤ 18, and larger n for small k.

Identity

Calls

count_running falling_step

Called by

factorial poly_derivative_k

sha256:d71751d0675a083c3cf834726b48dda06637c4be54d8937f35efdc6805a3737a

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.elementwise.permutations · Ω:permutations
Pins: sha256:8ec35594adafc18ab22e49691782517811d11df42cfa0da0dc89f6591ac334b2this graph and the 2 it reaches through calls
Evidence: sha256:fbee99eee3ab5824162d3f2fcc715e491a3ea1c4be4d422918c92411981d94bathe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.