What is the formula or piecewise function that gives how many moves [or half moves (plies)] are the DTMs if there are multiple checkmate variations (Depth to Mate ignoring 50-move rule) for a random $N$-piece chess position if any checkmate exists?
Also consider the positions that there is only one checkmate variation. And $N$ is at most 32. DTM is the number of moves for one side to checkmate the other [or half moves (plies)] while ensuring optimal play from White and Black.
Without neglecting ONLY THE MANDATORY FIDE draw rules and the DEAD POSITION rule.
Assume that the losing side tries to maximize DTM and the winning side tries to minimize DTM (Losing side fighting to delay checkmate, winning side trying to deliver the checkmate as quickly as possible) such that both sides play optimally starting from the position.