MathDB

A table of the type

~

(n_1, n_2, ... , n_m) ,\ n_1 \ge n_2 \ge ... \ge n_m

~

is defined in the following way:

~

n_1

~

squares are ordered horizontally one next to another, then

~

n_2

~

squares are ordered horizontally beneath the already ordered

~

n_1

~

squares. The procedure continues until a net composed of

~

n_1

~

squares in the first row,

~

n_2

~

in the second,

~

n_i

~

in the

~

i

-th row is obtained, such that there are totally

~

n=n_1+n_2+...+n_m

~

squares in the net. The ordered rows form a straight line on the left, as shown in the example. The obtained table is filled with the numbers from

~

1

~

till

~

n

~

in a way that the numbers in each row and column become greater from left to right and from top to bottom, respectively. An example of a table of the type

~

(5,4,2,1)

~

and one possible way of filling it is attached to the post. Find the number of ways the table of type

~

(4,3,2)

~

can be filled.

Problem Statement