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Category Archives: combinatorics
An Alternating Convolution Identity via SignReversing Involutions
This month’s post is on the combinatorial proof technique of signreversing involutions. It’s a really clever idea that can often be applied to identities that feature alternating sums. We’ll illustrate the technique on the following identity. . (Here, we use … Continue reading
Posted in binomial coefficients, combinatorics
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Minimum Number of Wagons to Connect Every City in Ticket to Ride – Europe
How many wagons would you need to create a route network in the board game Ticket to Ride – Europe so that every city is connected to every other city through the network? If you could do this you could potentially … Continue reading
Posted in combinatorial optimization, games, graph theory
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Counting Poker Hands
For this post I’m going to go through a classic exercise in combinatorics and probability; namely, proving that the standard ranking of poker hands is correct. First, here are the standard poker hands, in ranked order. Straight flush: Five cards of … Continue reading
Posted in combinatorics, probability
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Two Methods for Proving an Alternating Binomial Identity
Recently I gave an optional homework problem in which I asked students to prove the following binomial identity: . (Here, I’m using the Iverson bracket notation in which if P is true and if P is false.) I intended for students to … Continue reading
Posted in binomial coefficients, combinatorics
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A Proof of Dobinski’s Formula for Bell Numbers
Dobinski’s formula entails the following infinite series expression for the nth Bell number : In this post we’ll work through a proof of Dobinski’s formula. We’ll need four formulas: The Maclaurin series for : . The formula for converting normal powers to falling … Continue reading
Posted in Bell numbers, sequences and series, Stirling numbers
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Integrality of the Catalan Numbers via Kummer’s Theorem
Why is the nth Catalan number, , an integer? If you know one of its combinatorial interpretations, then the answer is clear, but how do you get integrality strictly from this formula? In this post I’m going to discuss how one can … Continue reading
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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