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Hockey-stick identity

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Pascal's triangle, rows 0 through 7. The hockey stick identity confirms, for example: for n=6, r=2: 1+3+6+10+15=35.

In combinatorial mathematics, the hockey-stick identity,[1] Christmas stocking identity,[2] boomerang identity, Fermat's identity or Chu's Theorem,[3] states that if are integers, then

The name stems from the graphical representation of the identity on Pascal's triangle: when the addends represented in the summation and the sum itself are highlighted, the shape revealed is vaguely reminiscent of those objects (see hockey stick, Christmas stocking).

Formulations

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Using sigma notation, the identity states

or equivalently, the mirror-image by the substitution :

Proofs

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Generating function proof

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Let . Then, by the partial sum formula for geometric series, we find that

.

Further, by the binomial theorem, we also find that

.

Note that this means the coefficient of in is given by .

Thus, the coefficient of in the left hand side of our first equation can be obtained by summing over the coefficients of from each term, which gives

Similarly, we find that the coefficient of on the right hand side is given by the coefficient of in , which is

Therefore, we can compare the coefficients of on each side of the equation to find that

Inductive and algebraic proofs

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The inductive and algebraic proofs both make use of Pascal's identity:

Inductive proof

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This identity can be proven by mathematical induction on .

Base case Let ;

Inductive step Suppose, for some ,

Then

Algebraic proof

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We use a telescoping argument to simplify the computation of the sum:

Combinatorial proofs

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Proof 1

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Imagine that we are distributing indistinguishable candies to distinguishable children. By a direct application of the stars and bars method, there are

ways to do this. Alternatively, we can first give candies to the oldest child so that we are essentially giving candies to kids and again, with stars and bars and double counting, we have

which simplifies to the desired result by taking and , and noticing that :

Proof 2

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We can form a committee of size from a group of people in

ways. Now we hand out the numbers to of the people. We can then divide our committee-forming process into exhaustive and disjoint cases based on the committee member with the lowest number, . Note that there are only people without numbers, meaning we must choose at least one person with a number in order to form a committee of people. In general, in case , person is on the committee and persons are not on the committee. The rest of the committee can then be chosen in

ways. Now we can sum the values of these disjoint cases, and using double counting, we obtain


See also

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References

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  1. ^ CH Jones (1996) Generalized Hockey Stick Identities and N-Dimensional Block Walking. Fibonacci Quarterly 34(3), 280-288.
  2. ^ W., Weisstein, Eric. "Christmas Stocking Theorem". mathworld.wolfram.com. Retrieved 2016-11-01.{{cite web}}: CS1 maint: multiple names: authors list (link)
  3. ^ Merris, Russell (2003). Combinatorics (2nd ed.). Hoboken, N.J.: Wiley-Interscience. p. 45. ISBN 0-471-45849-X. OCLC 53121765.
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