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A180632 Minimum length of a string of letters that contains every permutation of n letters as sub-strings, also known as length of the minimal super-permutation. 4
0, 1, 3, 9, 33, 153 (list; graph; refs; listen; history; text; internal format)



Obviously bounded below by n!+n-1 and above by 2(n!-(n-1)!)+1.

A better lower bound is n! + (n-1)! + (n-2)! + n-3 and a better upper bound is A007489(n).

Was conjectured to be equal to A007489, but it is now known that a(n) < A007489(n) for all n > 5. - Robin Houston, Aug 22 2014

Different from the minimal supersequence, in which each permutations of n letters can appear as a subsequence instead of a sub-string (i.e., with noncontiguous characters). Refer to A062714. - Maurizio De Leo, Mar 02 2015


D. Ashlock and J. Tillotson. Construction of small superpermutations and minimal injective superstrings. Congressus Numerantium, 93 (1993), 91-98.


Table of n, a(n) for n=0..5.

J. A. Barnett, Permutation Strings

Robin Houston, Tackling the Minimal Superpermutation Problem, arXiv:1408.5108, 2014.

Nathaniel Johnston, Non-uniqueness of minimal superpermutations, Discrete Math., 313 (2013), 1553-1557.

Nathaniel Johnston, The minimal superpermutation problem, 2013.

Nathaniel Johnston, All minimal superpermutations on five symbols have been found


For n = 1, 2, 3, 4, the (unique, up to relabeling the symbols) minimal words are:





For n = 5, there are exactly 8 distinct (up to relabeling the symbols) minimal words.

Comment from N. J. A. Sloane, Mar 27 2015: From the Houston (2014 arXiv) paper, a superpermutation of length 872 (not known to be minimal, but shorter than the old upper bound of 873):

1234561234516234512634512364513264513624513642513645213645123465123415 6234152634152364152346152341652341256341253641253461253416253412653412 3564123546123541623541263541236541326543126453162435162431562431652431 6254316245316425314625314265314256314253614253164523146523145623145263 1452361452316453216453126435126431526431256432156423154623154263154236 1542316542315642135642153624153621453621543621534621354621345621346521 3462513462153642156342165342163542163452163425163421564325164325614325 6413256431265432165432615342613542613452613425613426513426153246513246 5312463512463152463125463215463251463254163254613254631245632145632415 6324516324561324563124653214653241653246153264153261453261543265143625 1436521435621435261435216435214635214365124361524361254361245361243561 2436514235614235164235146235142635142365143265413625413652413562413526 41352461352416352413654213654123


Cf. A188428, A224986, A062714.

Sequence in context: A049425 A012584 A101899 * A009220 A007489 A201968

Adjacent sequences:  A180629 A180630 A180631 * A180633 A180634 A180635




Michael Hamm, Sep 13 2010


Edited and expanded by Nathaniel Johnston, Apr 22 2013

a(5) computed by Benjamin Chaffin and verified by Nathaniel Johnston, Aug 13 2014

Name edited by Maurizio De Leo, Mar 02 2015



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Last modified November 30 22:52 EST 2015. Contains 264677 sequences.