  In this paper we give a bijection between the class of permutations that can
be drawn on an X-shape and a certain set of permutations that appears in
[Knuth] in connection to sorting algorithms. A natural generalization of this
set leads us to the definition of almost-increasing permutations, which is a
one-parameter family of permutations that can be characterized in terms of
forbidden patterns. We find generating functions for almost-increasing
permutations by using their cycle structure to map them to colored Motzkin
paths. We also give refined enumerations with respect to the number of cycles,
fixed points, excedances, and inversions.
