[ Arrangements and Partitions ] $[$ Case Analysis $]$

Find the number of all Young diagrams of weight $s$, if
a) $s=4 ;$ b) $s=5$
c) $s=6 ;$ d) $s=7$.

For the definition of Young diagrams, see the reference manual.
List all the ways to break down the given weights into sums of natural addends, arranged in non-decreasing order.

a) $4=3+1=2+2=2+1+1=1+1+1+1$.

b) $5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1$.

c) $6=5+1=4+2=4+1+1=3+3=3+2+1=3+1+1+1=2+2+2=2+2+1+1=2+1+1+1+1=1+$ $1+1+1+1+1$

d) $7=6+1=5+2=5+1+1=4+3=4+2+1=4+1+1+1=3+3+1=3+2+2=3+2+1+1=3+1+1+$ $1+1=2+2+2+1=$

$=2+2+1+1+1=2+1+1+1+1+1=1+1+1+1+1+1+1$.

## Answer

a) 5 ; b) 7 ; c) 11 ; d) 15 diagrams.