Jumping through Lexicographic Permutations — Solution¶
Setter method: descending suffixes and factorial-sized lexicographic blocks. Codex (GPT-6) clarified cyclic semantics, supplied dynamic storage, proofs and validation, and added the Lehmer-digit alternative.
Jump from the end of a permutation block¶
After fixing a prefix, \(s\) remaining distinct elements have \(s!\) arrangements. Choices for the next element form consecutive equal-sized blocks in increasing value order.
Scan position \(i\) right to left, maintaining a descending suffix of length \(s=n-i\) . The current permutation is the last one in its prefix block. Suppose \(g\) suffix elements exceed \(a_i\) ; they occupy the first \(g\) positions of the descending suffix. The last permutation of the entire suffix \([i,n]\) is exactly \(g\cdot s!\) steps ahead.
If enough steps remain, subtract that distance and move \(a_i\) past those larger elements. The enlarged suffix is descending, so continue leftward. If \(g=0\) , the invariant already holds without any change.
Land inside a block¶
If fewer than \(g\cdot s!\) steps remain, spend one step reaching the first arrangement of the next first-element block. The remaining distance skips
additional blocks. Choose the \((q+1)\)-st smallest suffix value greater than \(a_i\) . Reverse the suffix into ascending order, swap in the chosen value, and insert the displaced smaller element leftward into its sorted position. The remaining distance becomes \((k-1)\bmod s!\) .
Now unrank within an ascending suffix: choose its \((\lfloor k/s!\rfloor+1)\)-st smallest value for the next position, then use the remainder for subsequent positions. Swaps and insertion keep the unused suffix sorted. Only sufficiently small factorials can have nonzero digits.
Here \(12!\le2^{31}-1<13!\) , so store factorials only through \(12!\) . A suffix longer than 12 has blocks larger than any remaining distance, hence quotient zero. After the descending permutation, one step wraps to the ascending permutation. Reduce \(k\) modulo \(n!\) when \(n\le12\) ; otherwise \(k<n!\) and at most one wrap occurs.
Correctness¶
The descending-suffix invariant locates the current permutation at a block end, making \(g\cdot s!\) the exact distance to the enlarged suffix's end. Full jumps cannot skip the requested destination. When a full jump is impossible, crossing the boundary and dividing into blocks selects the correct next value; the remainder is exactly the desired suffix rank. Sorted-suffix unranking realizes that rank position by position. The final wrap also follows the specified cycle, proving the result.
Complexity and Tradeoffs¶
Let \(K\) be the allowed maximum step count and \(D\) the smallest positive integer with \(D!>K\) ; here \(D=13\) . Scanning and at most one long suffix reversal and insertion cost \(O(n)\) ; short-tail work costs \(O(D^2)\) . Each case takes \(O(n+D^2)\) time and \(O(n+D)\) space. At most two calls preserve these bounds. Total input/output volume follows the aggregate length constraint.
Use std.cpp by default. The Fenwick alternative expresses ranks more directly but takes \(O(n\log n)\) and exceeds the measured 0.5-second limit on full stress cases. It remains a correct backup and cross-check.