Short Superpermutations over 8, 9, and 10

FrontierMath: Open Problems lists this problem with notability “Moderately interesting”, field “Combinatorics” and type “Construction - Family; Improved Bound”. Each problem has a bespoke verifier program, so a proposed answer can be checked automatically.

Verifiable task — Epoch prompt, verbatim

A superpermutation on n symbols is a sequence of integers, each in {1, ..., n}, that contains every one of the n! permutations of (1, 2, ..., n) as a contiguous (consecutive) subsequence. For example, the sequence

    1 2 1

is a superpermutation on 2 symbols, because its two length-2 windows, (1, 2) and (2, 1), are exactly the two permutations of (1, 2). Likewise

    1 2 3 1 2 1 3 2 1

is a superpermutation on 3 symbols of length 9: sliding a window of length 3 across it yields all six permutations of (1, 2, 3).

A central question is how short a superpermutation on n symbols can be. The minimum length is known exactly only for small n (it equals 1! + 2! + ... + n! for n <= 5: 1, 3, 9, 33, 153). For n >= 6 the minimum is unknown, and the best results are upper bounds from explicit constructions.

## Task

Find three short superpermutations:

  * seq8  : a superpermutation on 8 symbols of length at most 46204
  * seq9  : a superpermutation on 9 symbols of length at most 408965
  * seq10 : a superpermutation on 10 symbols of length at most 4037046

Each sequence must use only the symbols 1, ..., n (with n = 8, 9, 10 respectively) and must contain all n! permutations as consecutive windows. All three must satisfy their length bounds for the submission to be fully accepted; a submission in which only some of the sequences are valid earns partial credit.

## Submission

Write your answer to a single text file containing one labelled list per sequence:

    seq8: [1, 2, 3, ...]
    seq9: [1, 2, 3, ...]
    seq10: [1, 2, 3, ...]

Each list is an ordinary bracketed, comma-separated list of integers and may span multiple lines. Submit the absolute path of this file using the submit tool. Verification is exact: a sequence that misses even one permutation, uses a symbol outside its range, or exceeds its length bound will be rejected.

Epoch marks this problem solved (AI) on 2026-07-27. This is Epoch’s classification; the solution has not been independently checked for this record.

Epoch solution update

On July 26, 2026, Uku Raudvere posted a superpermutation over 8 smaller than the then-current upper bound to GitHub and to the Superpermutators Google group. Inspired by that post, on the same day, William Echols posted superpermutations over 9 and 10 smaller than the then-current upper bound, which were “developed in an interactive Codex session using OpenAI’s GPT-5.6-Sol at xhigh reasoning.” Discussion in the group suggests that this may generalize to an improved bound for all $n$.

GPT 5.6 Sol sourced the permutations over 8, 9, and 10 from these links and submitted them during a pre-release test of FM:OP done by Epoch on July 27, 2026, successfully passing the superpermutations verifier.

Since the superpermutations problem appears to have been solved by AI after we accepted the problem into the benchmark, we are recording this problem as solved on release. Once mathematicians have had a chance to digest these results, we may choose to ask AI for a further improvement to the bound.