# Motzkin Numbers and Their Geometrical Interpretation

Motzkin Numbers and Their Geometrical Interpretation

The top-left graphic is a three-dimensional plot of hybrid polynomials in two variables and ; the top-right blue plot is the surface cut by a plane perpendicular to the axis,; the bottom-left pink plot is the surface cut by a plane perpendicular to the axis. The representation of Motzkin numbers can be seen by moving the cutting planes to and , while keeping the order . The numbers correspond to the ordinates (that is, to the values of the polynomials read on the vertical axis) and are represented by a black dot on the blue and pink planes. The full sequence of Motzkin numbers can be explored by leaving and then moving the slider for .

x

y

y

x

x=1

y=1

q=1

x=y=q=1

n