2

Part of 2011 Kurschak Competition

Problems(1)

Equality between partition numbers

Source: Kürschák competition 2011 P2

x_1\leqslant x_2 \leqslant\dots

be a positive integer. Denote by

a(n)

the ways of expression

n=x_1+x_2+\dots

x_1\leqslant x_2 \leqslant\dots

are positive integers and

x_i+1

2

y_i

b(n)

the ways of expression

n=y_1+y_2+\dots

y_i

is a positive integer and

2y_i\leqslant y_{i+1}

i

a(n)=b(n)

inequalitiescombinatorics