20201124, 07:13  #1 
Nov 2020
2_{10} Posts 
Number with all permutations between any 2 digits
Lets say you are given a set of n consecutive digits , Ex {0,1,2,3}. You are required to form a number using the digits which abides by the following rule:
1.The number should cover all relations (ex 12 and 21) between any 2 digits from the set of numbers For the given set this number would be :0123130203210 or 0123021310320. I was able to get this number manually by trial and error. Is there a deterministic way of achieving this number for any n consecutive digits? The number will have all the 2 digit permutations only once. 0 > [1,2,3] will yield 01 , 02 , 03 1 > [0,2,3] will yield 10 , 12 , 13 2 > [0,1,3] will yield 20 , 21 , 23 3 > [0,1,2] will yield 30 , 31 , 32 We have to generate a number which will cover all the above relations. Last fiddled with by sticky on 20201124 at 08:04 
20201124, 10:14  #2 
Romulan Interpreter
Jun 2011
Thailand
2^{4}·571 Posts 
Why can't you just list all C(n,2) in a row lexicografic and add a zero at the end?
0102031213230 This is deterministic and correct (can you prove?). 
20201124, 13:24  #3 
"Viliam Furík"
Jul 2018
Martin, Slovakia
5^{2}×13 Posts 
I think what you are looking for is the Superpermutation page on Wikipedia.

20201124, 13:44  #4 
Feb 2017
Nowhere
10100_{8} Posts 
1) It occurred to me to wonder whether this might be a homework problem.
2) IMO this is not "superpermutations." For the problem as described, the the "list all transpositions" approach seems to be appropriate. A concept related, but not directly related to this problem, is that of a transitive group. 
20201124, 17:12  #5 
Nov 2020
2 Posts 
The answer is Eulerian path.
Algorithm is Hierholzers path detection. 
20201124, 17:21  #6 
"Viliam Furík"
Jul 2018
Martin, Slovakia
5^{2}·13 Posts 
Yes, but it's something like generalized superpermutations, with generalization being the fact that they consist of permutations of combinations of the symbols. So kind of like "Combinational superpermutations".
The example, 0123021310320, contains every permutation of every twoelement combination of the 4 elements  {0;1;2;3}. 
20201125, 03:58  #7 
"Matthew Anderson"
Dec 2010
Oregon, USA
2×317 Posts 
I didn't quite understand the question,
but I believe the Alternating group is relevant. Briefly it is the set of all even number of transpositions. see Wikipedia Alternating group. Regards, Matt 
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