# Tagged Questions

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### What is the motivation behind the study of pattern-avoiding permutations?

There is a ton of research on pattern-avoiding permutations (permutations that do not contain some designated permutation pattern). We're figuring out how to enumerate them, what random ones are ...
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### Iterations of Pascal's Identity

Pascal's Identity states that $n \choose k$ = $n-1 \choose k-1$ + $n-1 \choose k$ since if one element is separated from the rest we can claim that either it is chosen (resulting in $k-1$ elements ...
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### How many answers can be created using the elementary arithmetic operators?

If I gave you an amount of $n$ numbers, how many anwswer will you be able to create using the elementary arithmetic operators ($+, -, \times, /$)? These are the rules: All numbers ...
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### An unexpected application of non-trivial combinatorics

PROBLEM STATEMENT Given two finite sets $A$ and $B$, each containing $s \in \mathbb N$ elements, how many pairs of functions $f \colon A \rightarrow B$ and $g \colon B \rightarrow A$ are there, ...
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### Uses of the incidence matrix of a graph

The incidence matrix of a graph is a way to represent the graph. Why go through the trouble of creating this representation of a graph? In other words what are the applications of the incidence matrix ...
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### Exceptional books on real world applications of graph theory.

What are some exceptional graph theory books geared explicitly towards real-world applications? I would be interested in both general books on the subject (essentially surveys of applied graph ...
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### Blackjack 21 training

A friend of mine has to do An essay on this and I thought this was a great place to ask. In the movie: the main character is seen reading a lot of books which he calls simple arithmetic instead of ...
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### Combinations for Screen lock For iPhone / Android

I was wondering if there was a general for formula to calculate the combination of the password lock for the current smart phones The following is the condition We must use four nodes or more to ...
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### Applications of the number of spanning trees in graphs

Let $G$ be a simple graph and denote by $\tau(G)$ the number of spanning trees of $G$. There are many results related to $\tau(G)$ for certain types of graphs. For example one of the prettiest (to ...
The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any ...