One of my favourite examples of an “obvious” mathematical statement that is actually false is the “fact” that if \(R,S,T \subseteq \mathbb{R}^n\) are vector spaces then

\(\dim(R + S + T) = \dim(R) + \dim(S) + \dim(T) \\ {}_{{}_{{}_{.}}}\quad\quad\quad\quad\quad\quad\quad\quad – \dim(R\cap S) -\dim(R\cap T) -\dim(S\cap T) \\ {}_{{}_{{}_{.}}}\quad\quad\quad\quad\quad\quad\quad\quad+\dim(R\cap S\cap T).\)

The reason that the above statement seems so obvious is that the similar fact \(\dim(R + S) = \dim(R) + \dim(S) – \dim(R\cap S)\) does hold, so it’s very tempting to think  “inclusion-exclusion, yadda yadda, it’s simple enough to prove that it’s not worth writing down or working through the details”. However, it’s not true: a counterexample is provided by 3 distinct lines through the origin in \(\mathbb{R}^2\).

There is another problem that I’ve been thinking about for quite some time that is also “obvious”: the minimal superpermutation conjecture. This conjecture was so obvious, in fact, that it appeared as a question in a national programming contest in 1998. Well, last night Robin Houston posted a note on the arXiv showing that, despite being obvious, the conjecture is false [1].

Superpermutations

What is the shortest string that contains each permutation of “123” as a contiguous substring? It is straightforward to check that “123121321” contains each of “123”, “132”, “213”, “231”, “312”, and “321” as substrings (i.e., it is a superpermutation of 3 symbols), and it’s not difficult to argue (or use a computer search to show) that it is the shortest string with this property.

Well, we can repeat this question for any number of symbols. I won’t repeat all of the details (because I already wrote about the problem here), but there is a natural recursive construction that takes an (n-1)-symbol superpermutation of length L and spits out an n-symbol superpermutation of length L+n!. This immediately gives us an n-symbol superpermutation of length 1! + 2! + 3! + … + n! for all n. Importantly, it seemed like this construction was the best we could do: computer search verifies that these superpermutations are the smallest possible, and are even unique, for n ≤ 4.

Furthermore, it is not difficult to come up with some lower bounds on the length of superpermutations that seem to suggest that we have found the right answer. A trivial argument shows that an n-symbol superpermutation must have length at least (n-1) + n!, since we need n characters for the first permutation, and 1 additional character for each of the remaining n!-1  permutations. This argument can be refined to show that a superpermutation must actually have length at least (n-2) + (n-1)! + n!, since there is no way to pack the permutations tightly enough so that each one only uses 1 additional character (spend a few minutes trying to construct superpermutations by hand and you’ll see this for yourself). In fact, we can even refine this argument further (see a not-so-pretty proof sketch here) to show that n-symbol superpermutations must have length at least (n-3) + (n-2)! + (n-1)! + n!.

A-ha! A pattern has emerged – surely we can just keep refining this argument over and over again to eventually get a lower bound of 1! + 2! + 3! + … + n!, which shows that the superpermutations we already have are indeed minimal, right? Some variant of this line of thought seemed to be where almost everyone’s mind went when introduced to this problem, and it seemed fairly convincing: this argument is more or less contained within the answers when this question was posted on MathExchange and on StackOverflow (although the authors are usually careful to state that their method only appears to be optimal), and this problem was presented as a programming question in the 1998 Turkish Informatics Olympiad (see the resulting thread here). Furthermore, even on pages where this was acknowledged to be a difficult open problem, it was sometimes claimed that it had been proved for n ≤ 11 (example).

For the above reasons, it was a long time before I was even convinced that this problem was indeed unsolved – it seemed like people had solved this problem but just found it not worth the effort of writing up a full proof, or that people had found a simple way to tackle the problem for moderately large values of n like 10 or 11 that I couldn’t even dream of handling.

The Conjecture is False

It turns out that the minimal superpermutation conjecture is false for all n ≥ 6. That is, there exists a superpermutation of length strictly less than 1! + 2! + 3! + … + n! in all of these cases [1]. In particular, Robin Houston found the following 6-symbol superpermutation of length 872, which is smaller than the conjectured minimum length of 1! + 2! + … + 6! = 873:

12345612345162345126345123645132645136245136425136452136451234651234156234152634
15236415234615234165234125634125364125346125341625341265341235641235461235416235
41263541236541326543126453162435162431562431652431625431624531642531462531426531
42563142536142531645231465231456231452631452361452316453216453126435126431526431
25643215642315462315426315423615423165423156421356421536241536214536215436215346
21354621345621346521346251346215364215634216534216354216345216342516342156432516
43256143256413256431265432165432615342613542613452613425613426513426153246513246
53124635124631524631254632154632514632541632546132546312456321456324156324516324
56132456312465321465324165324615326415326145326154326514362514365214356214352614
35216435214635214365124361524361254361245361243561243651423561423516423514623514
263514236514326541362541365241356241352641352461352416352413654213654123

So not only are congratulations due to Robin for settling the conjecture, but a big “thank you” are due to him as well for (hopefully) convincing everyone that this problem is not as easy as it appears upon first glance.

References

1. R. Houston. Tackling the Minimal Superpermutation Problem. E-print: arXiv:1408.5108 [math.CO], 2014.

Recall from an earlier blog post that the minimal superpermutation problem asks for the shortest string on the symbols “1”, “2”, …, “n” that contains every permutation of those symbols as a contiguous substring. For example, “121” is a minimal superpermutation on the symbols “1” and “2”, since it contains both “12” and “21” as substrings, and there is no shorter string with this property.

Until now, the length of minimal superpermutations has only been known when n ≤ 4: they have length 1, 3, 9, and 33 in these cases, respectively. It has been conjectured that minimal superpermutations have length \(\sum_{k=1}^n k!\) for all n, and I am happy to announce that Ben Chaffin has proved this conjecture when n = 5. More specifically, he showed that minimal superpermutations in the n = 5 case have length 153, and there are exactly 8 such superpermutations (previously, it was only known that minimal superpermutations have either length 152 or 153 in this case, and there are at least 2 superpermutations of length 153).

The Eight Minimal Superpermutations

The eight superpermutations that Ben found are available here (they’re too large to include in the body of this post). Notice that the first superpermutation is the well-known “easy-to-construct” superpermutation described here, and the second superpermutation is the one that was found in [1]. The other six superpermutations are new.

One really interesting thing about the six new superpermutations is that they are the first known minimal superpermutations to break the “gap pattern” that previously-known constructions have. To explain what I mean by this, consider the minimal superpermutation “123121321” on three symbols. We can think about generating this superpermutation greedily: we start with “123”, then we append the character “1” to add the permutation “231” to the string, and then we append the character “2” to add the permutation “312” to the string. But now we are stuck: we have “12312”, and there is no way to append just one character to this string in such a way as to add another permutation to it: we have to append the two characters “13” to get the new permutation “213”.

This phenomenon seemed to be fairly general: in all known small superpermutations on n symbols, there was always a point (approximately halfway through the superpermutation) where n-2 consecutive characters were “wasted”: they did not add any new permutations themselves, but only “prepared” the next symbol to add a new permutation.

However, none of the six new minimal superpermutations have this property: they all never have more than 2 consecutive “wasted” characters, whereas the two previously-known superpermutations each have a run of n-2 = 3 consecutive “wasted” characters. Thus these six new superpermutations are really quite different from any superpermutations that we currently know and love.

How They Were Found

The idea of Ben’s search is to do a depth-first search on the placement of the “wasted” characters (recall that “wasted” characters were defined and discussed in the previous section). Since the shortest known superpermutation on 5 symbols has length 153, and there are 120 permutations of 5 symbols, and the first n-1 = 4 characters of the superpermutation must be wasted, we are left with the problem of trying to place 153 – 120 – 4 = 29 wasted characters. If we can find a superpermutation with only 28 wasted characters (other than the initial 4), then we’ve found a superpermutation of length 152; if we really need all 29 wasted characters, then minimal superpermutations have length 153.

So now we do the depth-first search:

• Find (via brute-force) the maximum number of permutations that we can fit in a string if we are allowed only 1 wasted character: the answer is 10 permutations (for example, the string “123451234152341” does the job).
• Now find the maximum number of permutations that we can fit in a string if we are allowed 2 wasted characters. To speed up the search, once we have found a string that contains some number (call it p) of permutations, we can ignore all other strings that use a wasted character before p-10 permutations, since we know from the previous bullet point that the second wasted character can add at most 10 more permutations, for a total of (p-10)+10 = p permutations.
• We now repeat this process for higher and higher numbers of wasted characters: we find the maximum number of permutations that we can fit in a string with 3 wasted characters, using the results from the previous two bullets to speed up the search by ignoring strings that place 1 or 2 wasted characters too early.
• Etc.

The results of this computation are summarized in the following table:

Wasted characters	Maximum # of permutations
0	5
1	10
2	15
3	20
4	23
5	28
6	33
7	36
8	41
9	46
10	49
11	53
12	58
13	62
14	66
15	70
16	74
17	79
18	83
19	87
20	92
21	96
22	99
23	103
24	107
25	111
26	114
27	116
28	118
29	120

As we can see, it is not possible to place all 120 permutations in a string with 28 or fewer wasted characters, which proves that there is no superpermutation of length 152 in the n = 5 case. C code that computes the values in the above table is available here.

Update [August 18, 2014]: Robin Houston has found a superpermutation on 6 symbols of length 873 (i.e., the conjectured minimal length) with the interesting property that it never has more than one consecutive wasted character! The superpermutation is available here.

IMPORTANT UPDATE [August 22, 2014]: Robin Houston has gone one step further and disproved the minimal superpermutation conjecture for all n ≥ 6. See here.

References

1. N. Johnston. Non-uniqueness of minimal superpermutations. Discrete Mathematics, 313:1553–1557, 2013.