MathDB

Let

n

and

k

be positive integers. An

n

-tuple

(a_1, a_2,\ldots , a_n)

is called a permutation if every number from the set

\{1, 2, . . . , n\}

occurs in it exactly once. For a permutation

(p_1, p_2, . . . , p_n)

, we define its

k

-mutation to be the

n

-tuple

(p_1 + p_{1+k}, p_2 + p_{2+k}, . . . , p_n + p_{n+k}),

where indices are taken modulo

n

. Find all pairs

(n, k)

such that every two distinct permutations have distinct

k

-mutations.
Remark: For example, when

(n, k) = (4, 2)

, the

2

-mutation of

(1, 2, 4, 3)

(1 + 4, 2 + 3, 4 + 1, 3 + 2) = (5, 5, 5, 5)

.
Proposed by Borna Šimić

Problem Statement