Permutations are among the most basic elements of discrete mathematics. They are used to represent discrete groups of transformations, and in particular play a key role in group theory, the mathematical study of symmetry. Permutations and groups are important in many aspects of life. We all live on a giant sphere (the Earth) whose rotations are described by the group SO(3) (the special orthogonal group in 3 dimensions). On the micro-scale, the Hungarian-American physicist Eugene Wigner (November 17, 1902–January 1, 1995), who received a share of the Nobel Prize in Physics in 1963, discovered the “electron permutation group”, one of many applications of permutation groups to quantum mechanics.

Permutations deserve a full treatment in Wolfram|Alpha. Since *Mathematica* 8 provides new functionality to work with permutations using both list and cyclic representations, we now have a powerful new way of working with permutations using natural language.

Let’s first define permutations, then discuss how to work with them in Wolfram|Alpha. A permutation of a set *X* is a bijective (one-to-one and onto) mapping of *X* to itself. There is a convenient way of specifying a permutation *α* of a finite set of *n* elements: write down the numbers 1, 2, …, *n* in a row and write down their images under *α* in a row beneath:

Last week we showed you some of the ways Wolfram|Alpha can display flight information. Now you can take that functionality with you wherever you go with the Wolfram Flight Information Reference App for iOS.

Creating strong passwords is an important part of online security. Now it is even easier to create strong passwords and check the strength of existing passwords with the Wolfram Password Generator Reference App for iOS.

At one time or another, we’ve all looked at a jet flying high overhead and thought “I wonder where they’re headed?” Actually answering that question probably seemed impossible before—but if you’re a user in the United States, Wolfram|Alpha can now help you answer that question and many more interesting queries about commercial and other flights.

Try the simple query “flights overhead” and you’ll get information on aircraft that should be visible to you, assuming a clear sky and unobstructed view. If you’re on a location-aware mobile device, the results should be based on your precise latitude and longitude—otherwise, Wolfram|Alpha will use the best available location information from your browser. Also note that hovering over an individual plane in the sky map will produce a tooltip with the airline and flight number:

Helping educators utilize Wolfram|Alpha in the classroom to enhance their lessons is one of our missions, and we love to learn about the creative ways teachers use Wolfram|Alpha.

One such example is Matt Arnold, a STEM (science, technology, engineering, and mathematics) teacher at Skiles Test Elementary in Indianapolis who started a Wolfram Math Club. The club consists of seven sixth-graders who meet twice a week to complete projects utilizing Wolfram|Alpha.

We are happy to announce that we released two new entries into our line of Wolfram|Alpha-powered iOS apps: the Wolfram Investment Calculator Reference App and the Wolfram Gaming Odds Reference App. The Wolfram Investment Calculator Reference App was built specifically to help users get the most from their investments, and the Wolfram Gaming Odds Reference App provides the probabilities and odds of winning many popular card and lottery games.

This Sunday, over 40,000 runners will take part in the New York City Marathon. A great amount of preparation and planning goes into accommodating such a large group of athletes, as well as the fans cheering on the sidelines. An article in *The New York Times* described the numbers behind planning such a huge race, so we decided to plug those numbers into Wolfram|Alpha and see what interesting things we could compute.

According to the article, over 43,000 athletes took part in the 26.2-mile race last year. Typing “running 26.2 miles” into Wolfram|Alpha allows you to enter your pace, gender, and body weight, then receive data based on your inputs, such as calories burned and oxygen consumed.