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 ($$${1}, {2}, \ldots, {N}$$$), which is the only permutation with zero inversions, is assigned the identifier $$$1$$$, while the reverse identity permutation ($$${N}, {N-1}, \ldots, {1}$$$), 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 $$$\pi$$$ with $$$N$$$ elements, its number of inversions $$$\mathop{\mathrm{inv}}(\pi)$$$ is defined as $$$$$$\mathop{\mathrm{inv}}(\pi) = \left| \{ (i, j) : 1 \leq i \lt j \leq N \land \pi(i) \gt \pi(j) \} \right| \enspace$$$$$$