@Nathaniel

Thank you! Perhaps, a search could be set up to find a 6-superpermutation of length 872, which contains 123456 twice. Due to symmetry, if we find such a superpermutation, it will generate a family of 720 solutions with all possible duplicates. On the other hand, if the search fails then there are no 6-superpermutations of length 872 with duplicates.

By the way, I just checked: Robin’s superpermutation doesn’t contain repetitions of non-permutations either. That is, all contiguous substrings of length 6 are distinct.

]]>Curiously, my program found 147 positions within the sequence that do not correspond to permutations (e.g., 123451 starting on 7th position). This means that there are 872 (all positions) – 720 (correct permutations) – 147 (not permutations) – 5 (beginning of the sequence) = 0 repetitions of valid permutations. That is, permutations are never repeated. It is not obvious for me if this must hold in all shortest superpermutations. Robin, do you know if this is always the case?

]]>– The operator-Schmidt coefficients are uniquely determined by .

– If all of the operator-Schmidt coefficients are distinct, then the operators and in the operator-Schmidt decomposition are uniquely determined as well (up to multiplication by a complex phase ).

– If there are degenerate (i.e., repeated) operator-Schmidt coefficients, then we can replace the ‘s by any linear combination of the ‘s corresponding to the same operator-Schmidt coefficient.

]]>