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Need help with this proof using Ferrers' graph or otherwise.

Show that the number of partitions of $r+k$ into $k$ parts is equal to

  1. The number of partitions of $r + {k+1 \choose 2}$ into $ k $ distinct parts
  2. The number of partitions of $r$ into parts of size at most $k$
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(i) Take the diagram of an arbitrary partition of $r+k$ into $k$ parts, then add $0$ to the smallest part, add $1$ to the second smallest part, add $2$ to the third, $\ldots$ add $(k-1)$ to the largest part and what have you got?

(ii) Take the diagram of an arbitrary partition of of $r+k$ into $k$ parts, then subtract $1$ from each part and what have you got? Now reflect it so rows become columns and columns become rows and what have you got?

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  • $\begingroup$ Very nice hint. $\endgroup$ – Brian M. Scott Oct 13 '12 at 4:11

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