Integer partition of n into k parts recurrence

I was learning integer partition of a number n into k parts(with minimum 1 in each part) and came across this recurrence :

part(n,k) = part(n-1,k-1) + part(n-k,k)

But, I cannot understand the logic behind this recurrence. Can someone please help me visualize this recurrence?

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Let’s consider an example: $n=8$ and $k=3$:

\begin{align*}
\\
P(n,k)&=P(n-1,k-1)+P(n-k,k)&\\
\\
P(8,3)&=P(7,2)+P(5,3)&\\
\\
\end{align*}



\begin{array}{rlrlrl}
Note that the partitions of $P(8,3)$ that have smallest integer part equal to one correspond to the integer partitions of $P(7,2)$ whereas the partitions with smallest integer part $>1$ correspond to the partitions of $P(5,3)$: