C - Previous Permutation
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C - Previous Permutation
Score : $300$ points
Problem Statement
You are given a permutation $P = (P_1, \dots, P_N)$ of $(1, \dots, N)$, where $(P_1, \dots, P_N) \neq (1, \dots, N)$.
Assume that $P$ is the $K$-th lexicographically smallest among all permutations of $(1 \dots, N)$. Find the $(K-1)$-th lexicographically smallest permutation.
What are permutations?
A permutation of $(1, \dots, N)$ is an arrangement of $(1, \dots, N)$ into a sequence.
What is lexicographical order?
For sequences of length $N$, $A = (A_1, \dots, A_N)$ and $B = (B_1, \dots, B_N)$, $A$ is said to be strictly lexicographically smaller than $B$ if and only if there is an integer $1 \leq i \leq N$ that satisfies both of the following.
- $(A_{1},\ldots,A_{i-1}) = (B_1,\ldots,B_{i-1}).$
- $A_i < B_i$.
Constraints
- $2 \leq N \leq 100$
- $1 \leq P_i \leq N \, (1 \leq i \leq N)$
- $P_i \neq P_j \, (i \neq j)$
- $(P_1, \dots, P_N) \neq (1, \dots, N)$
- All values in the input are integers.
Input
The input is given from Standard Input in the following format:
Output
Let $Q = (Q_1, \dots, Q_N)$ be the sought permutation. Print $Q_1, \dots, Q_N$ in a single line in this order, separated by spaces.
3
3 1 2
2 3 1
Here are the permutations of $(1, 2, 3)$ in ascending lexicographical order.
- $(1, 2, 3)$
- $(1, 3, 2)$
- $(2, 1, 3)$
- $(2, 3, 1)$
- $(3, 1, 2)$
- $(3, 2, 1)$
Therefore, $P = (3, 1, 2)$ is the fifth smallest, so the sought permutation, which is the fourth smallest $(5 - 1 = 4)$, is $(2, 3, 1)$.
10
9 8 6 5 10 3 1 2 4 7
9 8 6 5 10 2 7 4 3 1