In the integer coordinate plane we consider those lattice
paths whose step types consists of (1,1), (1,-1), and
perhaps some horizontal steps. For those paths running from
(0,0) to (n+2,0) and remaining strictly above the x-axis
elsewhere, we will define a zeroth moment(the cardinality),
a first moment (the total area), and a second moment, in
terms of the lattice points traced by its paths. We will
define a bijection relating these moments to the
cardinalities of sets of selected lattice points on the
unrestricted paths running from (0,0) to (n,0). This bijection
acts by cutting the elevated paths and then shuffling the
resulting subpaths by a fixed permutation. For cardinality,
this ``cut shuffling'' is a variant of the well-known cycle
lemma.
