These little guys are like the cooler cousins of factorials (which you probably remember from high school math class) and they come in handy when dealing with combinatorics problems.

So, what exactly are these Stirling numbers? Well, let’s start by defining them:

Stirling number of the second kind, denoted by S(n,k), is a coefficient that appears in the expansion of (x + y) when you take out all the terms with x and y raised to powers greater than n. In other words, it’s like a fancy way of saying “how many ways can we choose k things from n options without repeating?”

Now, if that sounds confusing, don’t worry here’s an example: let’s say you have 5 candies and want to give your friend 3. How many different combinations can you make? Well, using Stirling numbers of the second kind, we get:

S(5,3) = (5 choose 3) / (3 choose 3) = 10

So there are 10 possible ways to give your friend 3 candies out of a total of 5. Pretty cool, right? But what if you want to find the number of ways to choose k things from n options with repetition allowed? Well, that’s where Stirling numbers of the first kind come in but we won’t talk about those today because they’re not as fun (sorry guys).

Anyway, let’s get back to our main topic: explicit formula for Stirling numbers of the second kind. As you might have guessed from the title, this is a fancy way of saying “there’s actually an easy-to-remember equation that can help us calculate these numbers”. And it goes like this:

S(n,k) = 1/k! * sum((-1)^i*(i+k-1 choose k-1)*x^i / (x-1)^(i+1)) from i=0 to n-k

Now, if you’re like me and don’t feel comfortable with all these fancy symbols and variables, let’s break it down:

– S(n,k) is the Stirling number of the second kind that we’ve been talking about. It represents the coefficient in the expansion of (x + y) when you take out all the terms with x and y raised to powers greater than n.

– k! stands for factorial it means multiply all the numbers from 1 to k together. For example, 5! is equal to 5 * 4 * 3 * 2 * 1 = 120.

– (i+k-1 choose k-1) is a combination function that tells us how many ways we can choose k things from i options without repeating. For example, if you have 5 candies and want to give your friend 3, the number of possible combinations would be:

(5 choose 3) = (5 * 4 * 3) / ((3 * 2 * 1)) = 10

– (-1)^i is a shorthand way of writing -1 to the power of i. For example, (-1)^3 would be equal to -1 * -1 * -1 = -1.

– x^i represents x raised to the power of i (so if x=2 and i=3, then x^i is 8).

– /(x-1)^(i+1) means divide by (x-1) raised to the power of (i+1). For example, if x=5 and i=3, then (x-1)^(i+1) would be equal to (5-1)^4 = 625.

It might look a bit intimidating at first glance, but once you break it down into smaller parts, it’s actually pretty easy to understand. And who knows? Maybe one day you’ll use this knowledge to impress your friends with some fancy math tricks (or just to win at trivia night).