# Cycles from patterns in a permutation matrix

• I
Gold Member
TL;DR Summary
Beyond fixed points and transpositions, is there an easy way to spot the subset of rows in a permutation matrix that indicate a cycle ?
In a permutation matrix (the identity matrix with rows possibly rearranged), it is easy to spot those rows which will indicate a fixed point -- the one on the diagonal -- and to spot the pairs of rows that will indicate a transposition: a pair of ones on a backward diagonal, i.e., where the row+column subscripts are a constant -- and of course one can find cycles by multiplying a vector and tracing around what goes to what. But is there some quicker way to spot the collections of rows that will correspond to a cycle?

Mentor
2022 Award
The cycle decomposition of a permutation should be ##O(n)##. How could you hope to find something faster? You still have to read those ##n## entries.