What is the length of the shortest sequence in [n]* containing, as a (consecutive) subword, each permutation of [n]

Definitions and mathematical statement

For an integer $n\ge1$, a superpermutation is a finite word $w=w_1\cdots w_L$ over the alphabet $[n]=\{1,\ldots,n\}$ such that every permutation $\sigma\in S_n$ occurs as a consecutive length-$n$ block. Precisely, for each $\sigma$ there is an index $1\le j\le L-n+1$ with \[ (w_j,\ldots,w_{j+n-1}) =(\sigma(1),\ldots,\sigma(n)). \] The word may repeat symbols and may contain blocks that are not permutations. Different required blocks may overlap.

Let $s(n)$ be the least possible length $L$ of such a word. Determine $s(n)$, or establish asymptotically sharp bounds for it as the alphabet size grows. Concatenating all permutations gives the elementary upper bound $s(n)\le n\,n!$, whereas counting available starting positions gives $s(n)\ge n!+n-1$.

The word is linear: an occurrence cannot wrap from its final letter back to its first. This convention differs from cyclic superpermutations. It also differs from superpatterns, where only relative order in a possibly nonconsecutive subsequence is required. Here every letter and its position in the block must agree exactly with the prescribed permutation. The optimization is over all words, without imposing a recursive construction, symmetry, or a particular order in which permutations first appear.

Short English statement

Determine the shortest length of a word containing every permutation of n symbols as a consecutive block.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3100036 (AMR-030-0036), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cooper - Combinatorial Problems I Like (2020 snapshot), Permutations, source-order bullet 36. Original source indexed by AMR-030-0036. https://people.math.sc.edu/cooper/combprob.html.