Inversion Insight

In MathemIsland, the wildlife is very diverse. There are roots, trees, leaves, pigeons – everything you’d find in a math book. And everywhere you look, there is a permutation.

ICPC University has devised a systematic way to catalog these permutations. Specifically, because inversions are of utmost importance in studying wildlife genetics, ICPC University has decided to sort all $N!$ permutations of the integers from $1$ to $N$: first by the number of inversions and, in the case of a tie, by lexicographic order. This approach uniquely identifies each permutation by an integer from $1$ to $N!$, indicating its position in the sorted list of all $N!$ permutations.

Thus, the identity permutation $(1, 2, \dots , N)$, which is the only permutation with zero inversions, is assigned the identifier $1$, while the reverse identity permutation $(N, N - 1, \dots , 1)$, which is the only one with the maximum number of inversions, is assigned the identifier $N!$.

As part of the team implementing the ICPC University database, your task is to retrieve a specific permutation based on its identifier. Write a program that, given two integers $N$ and $K$, outputs the permutation of the integers from $1$ to $N$ corresponding to identifier $K$.

Remember that the number of inversions in a permutation is the number of pairs of elements that are out of their natural order. That is, for a permutation $π$ with $N$ elements, its number of inversions $\text{inv}(π)$ is defined as

$$\text{inv}(π) = |\{(i, j) : 1 ≤ i < j ≤ N ∧ π(i) > π(j)\}|$$

The input consists of a single line that contains two integers $N$ ($1 ≤ N ≤ 2 \cdot 10^5 $) and $K$ ($1 ≤ K ≤ \min(N!, 4 \cdot 10^{18})$).

Output a single line with $N$ integers, describing the $K$-th permutation of the integers from $1$ to $N$, considering permutations sorted according to the university’s criteria.