Friday, 28 November 2014

combinatorics - Combinatorial identity: summation of stars and bars

I noticed that the following identity for a summation of stars and bars held for specific k but I was wondering if someone could provide a general proof via combinatorics or algebraic manipulation. I wouldn't be surprised if this is a known result; it looks very similar to the Hockey Stick identity.



\sum_{i=0}^k {d+i-1 \choose d-1} = {d+k \choose k}



The left can be immediately rewritten as \sum_{i=0}^k {d+i-1 \choose i} if it helps inspire intuition.

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