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question

A Skolem sequence of order nn is a sequence (s1,s2,,s2n)\left(s_{1}, s_{2}, \ldots, s_{2 n}\right) of 2n2 n integers satisfying the conditions: i) for every kk in {1,2,3,,n}\{1,2,3, \ldots, n\}, there exist exactly two elements sis_{i} and sjs_{j} with si=sj=ks_{i}=s_{j}=k, and ii) if si=sj=ks_{i}=s_{j}=k with i<ji<j, then ji=kj-i=k. For example, (4,2,3,2,4,3,1,1)(4,2,3,2,4,3,1,1) is a Skolem sequence of order 4. List all Skolem sequences of order 4.
Plain-text mathematical notation (without MathML)
A Skolem sequence of order n is a sequence (s₁,s₂,…,s_(2n)) of 2n integers satisfying the conditions:

i) for every k in {1,2,3,…,n}, there exist exactly two elements s_(i) and s_(j) with s_(i)=s_(j)=k, and

ii) if s_(i)=s_(j)=k with i<j, then j−i=k.


For example, (4,2,3,2,4,3,1,1) is a Skolem sequence of order 4.
List all Skolem sequences of order 4.
Original LaTeX notation
A Skolem sequence of order $n$ is a sequence $\left(s_{1}, s_{2}, \ldots, s_{2 n}\right)$ of $2 n$ integers satisfying the conditions:

i) for every $k$ in $\{1,2,3, \ldots, n\}$, there exist exactly two elements $s_{i}$ and $s_{j}$ with $s_{i}=s_{j}=k$, and

ii) if $s_{i}=s_{j}=k$ with $i<j$, then $j-i=k$.


For example, $(4,2,3,2,4,3,1,1)$ is a Skolem sequence of order 4.
List all Skolem sequences of order 4.

answer type

Tuple

is multiple answer

true

language

English

question type

Open-ended

subject

Math

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