# Category Archives: combinatorics

## Six Proofs of a Binomial Identity

I’m a big fan of proving an identity in multiple ways, as I think each perspective gives additional insight into why the identity is true.  In this post we’ll work through six different proofs of the binomial identity . 1. … Continue reading

## An Alternating Convolution Identity via Sign-Reversing Involutions

This month’s post is on the combinatorial proof technique of sign-reversing 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

## 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

## 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

## 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

## 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

## 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 Odd-Indexed 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