Recall that pk (n) is the number of partitions of n with k non-zero parts. Show...

  

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Recall that pk (n) is the number of partitions of n with k non-zero parts. Show that for any positive integers n, k we have ·+Pk (n)=Pk (n + k) P₁(n) + P2(n) + P3(N) + ···· by considering the operation of removing the first column from a Young diagram of a partition of n + k with k parts. Recall that pk (n) is the number of partitions of n with k non-zero parts. Show that for any positive integers n, k we have ·+Pk (n)=Pk (n + k) P₁(n) + P2(n) + P3(N) + ···· by considering the operation of removing the first column from a Young diagram of a partition of n + k with k parts.