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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
.
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