Lexicographic Permutations

Problem #90

Tags: popular-algorithm puzzle combinatorics

Who solved this?

Friar Geeglo was investigating wine cellars of CodeAbbey when he suddenly found something interesting.

Behind huge moss-covered barrel in the dark corner there was a narrow passage. Peeking there friar Geeglo discovered a small door, obviously leading to storage of long-forgotten delicious drinks!

Door has a whimsical lock - there are metal pads with letters A B C D E F G H I J K L which can be moved so their order is changed. It is clear - one just should arrange them all properly (i.e. guess the right permutation) - and the door magically opens.

Below small inscription was scratched I've tried first NN...N permutations in lexicographic order left by some unknown monk.

Brother Geeglo wants to proceed with the attempts of unlocking the door. However he do not want to check variants already tried by his predecessor (this can save him several years).

You are to help him find out which permutation he should start from.

For example, if there were only 4 letters, then 24 permutations exist and if placed in order they'll look like:

 0 ABCD     1 ABDC     2 ACBD     3 ACDB     4 ADBC     5 ADCB
 6 BACD     7 BADC     8 BCAD     9 BCDA    10 BDAC    11 BDCA
12 CABD    13 CADB    14 CBAD    15 CBDA    16 CDAB    17 CDBA
18 DABC    19 DACB    20 DBAC    21 DBCA    22 DCAB    23 DCBA

And if Geeglo is told that first 9 of them were already tried, he should proceed from BCDA.

Input data contain the number of test-cases in the first line.
Next lines contain a single value each - the number of permutations tried by unknown monk.
Answer should contain the same amount of letter permutations - ones with which Geeglo should start in each case.

Example:

input data:
3
0
2
10531100

answer:
ABCDEFGHIJKL ABCDEFGHIKJL ADLBEHICKGFJ

Note: You can find several algorithms for generating permutations in wikipedia article

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