1008 Permutations
Exercise
1.Find the one-billionth permutation in the universal sequence.2.The inverse permutation of (5,1,3,7,4,8,2,6) is (2,7,3,5,1,8,4,6). What
is its position number in the universal sequence?3.Let(a 1 ,..., (^1) n)be a permutation of(1, 2 ,...,n).
(i)
∑
cyclic|ai−ai+1|≥^2 n−^2.
(ii) For how many distinct permutations of(1, 2 ,...,n)does equal-
ity hold? Answer:n· 2 n.Project: Nice odd integers
An odd integernis said to be nice if and only if there is a permutation
(a 1 ,a 2 ,...,an)of(1, 2 ,...,n)such that the sums
a 1 −a 2 +···−an− 1 +an,
a 2 −a 3 +···−an+a 1 ,
..
.
an−a 1 +···−an− 2 +an− 1are all positive. Find all nice integers.