  In this note, we consider the problem of finding a step-by-step
transformation between two longest increasing subsequences in a sequence,
namely Longest Increasing Subsequence Reconfiguration. We give a
polynomial-time algorithm for deciding whether there is a reconfiguration
sequence between two longest increasing subsequences in a sequence. This
implies that Independent Set Reconfiguration and Token Sliding are
polynomial-time solvable on permutation graphs, provided that the input two
independent sets are largest among all independent sets in the input graph. We
also consider a special case, where the underlying permutation graph of an
input sequence is bipartite. In this case, we give a polynomial-time algorithm
for finding a shortest reconfiguration sequence (if it exists).
