Exact Counting and Uniform Sampling of Pressing Sequences on Paths and Cycles

We give an exact polynomial-time algorithm for counting successful pressing sequences of every labeled bicolored graph whose underlying graph has maximum degree at most two. Paths, cycles, arbitrary colorings, singular matrices, full-length sequences and disconnected unions are included. A determinant-gated last-vertex recurrence on path intervals, together with nullity bounds, yields O(n^3) arithmetic operations on O(1+n log(n+1))-bit integers. Exact uniform sampling follows from deterministic unranking in O(n^2) arithmetic operations after a uniform integer draw; fair-bit sampling has expected polynomial bit time. This answers the full-length path-counting alternative in Miklos, Dagstuhl Reports 13(3) (2023), section 4.8. General-graph hardness/approximation questions in AMR-030-0046 and rapid mixing of the earlier local-edit chain remain unresolved here. AI-assisted, self-audited preprint; no independent review, formal verification or absolute-priority certification is claimed.