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Category Archives: combinatorics
Tiling Proofs for the Sum of Even or OddIndexed Binomial Coefficients
Two basic identities for the binomial coefficients are contained in the equation . There are multiple ways to prove these identities (even if you restrict yourself to combinatorial proofs). In this post I’m going to discuss a tiling or … Continue reading
Posted in binomial coefficients, combinatorics
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Combinatorial Proofs of Two HagenRothe Identities in Concrete Mathematics
There are two binomial coefficient identities in Concrete Mathematics that have bothered me for years: They look they should have combinatorial proofs, but I couldn’t for the life of me think of what those might be. (In Concrete Mathematics they are … Continue reading
Posted in combinatorics, lattice paths
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Vandermonde’s Identity from the Generalized Product Rule
Vandermonde’s Identity, is one of the more famous identities involving the binomial coefficients. A standard way to prove it is with the following combinatorial argument. How many ways are there to choose a committee of size r from a group of n men … Continue reading
Posted in binomial coefficients, calculus, combinatorics
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Combinatorial Proofs for the Binomial and Alternating Binomial Power Sums
In this post I give combinatorial proofs of formulas for the binomial and alternating binomial power sums: and Here’s the first. Identity 1. . Proof. Both sides count the number of functions from to subsets of . For the left … Continue reading
Posted in binomial coefficients, combinatorics, Stirling numbers
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LogConvexity of the Bell Numbers
A sequence is said to be logconvex if, for all , . In this post I give a short, combinatorialflavored proof that the Bell numbers are logconvex. The argument is from my answer here on Math.SE. The combinatorial interpretation of the Bell … Continue reading
A Combinatorial Proof for the Power Sum
Probably the most common formula for the power sum is the one involving binomial coefficients and Bernoulli numbers , sometimes called Faulhaber’s formula: Historically, this, or a variant of it, was the first general formula for the power sum. There … Continue reading
Yet Another Nice Proof of the Derangements Formula
A couple of posts from a few years back give proofs of the formula for , the number of derangements on n elements. (A derangement is a permutation with no fixed points.) Both proofs avoid the use of the principle of inclusionexclusion, which … Continue reading
Posted in combinatorics, derangements, permutations
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