Motzkin Numbers and Their Geometrical Interpretation

bound for y and x

3

bound for z

30

cutting plane position y

1

cutting plane position x

1

polynomial degree n

3

integer order q

1

m

3

=

(1)

P

3

(1, 1) = 4

(1)

P

3

(x, y) = 6

3

x

6

+

xy

2

The top-left graphic is a three-dimensional plot of hybrid polynomials in two variables

x

and

y

; the top-right blue plot is the surface cut by a plane perpendicular to the

y

axis,; the bottom-left pink plot is the surface cut by a plane perpendicular to the

x

axis. The representation of Motzkin numbers can be seen by moving the cutting planes to

x=1

and

y=1

, while keeping the order

q=1

. 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

x=y=q=1

and then moving the slider for

n

.