MathDB

By a partition

\pi

of an integer

n\ge 1

, we mean here a representation of

n

as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if

n=4

, then the partitions

\pi

are

1+1+1+1

1+1+2

1+3, 2+2

, and

4

).
For any partition

\pi

, define

A(\pi)

to be the number of

1

's which appear in

\pi

, and define

B(\pi)

to be the number of distinct integers which appear in

\pi

. (E.g., if

n=13

and

\pi

is the partition

1+1+2+2+2+5

, then

A(\pi)=2

and

B(\pi) = 3

).
Prove that, for any fixed

n

, the sum of

A(\pi)

over all partitions of

\pi

n

is equal to the sum of

B(\pi)

over all partitions of

\pi

n

Problem Statement