It is well known that permutation puzzles like Loyd's 15-puzzle and the Rubik's cube each have an associated group. In most cases this group is the set of permutations of the elements of the puzzle which are accessible via sequences of legal puzzle moves (no fair removing stickers!) but in the case of the 15-puzzle we only consider those configurations in which the blank is in a given, fixed, position. Generally speaking, people design these puzzles for fun, and only later is the group of the puzzle determined. In an article in the July 2008 issue of Scientific American, Igor Kriz and Paul Siegel turned this paradigm around: instead of designing puzzles and then determining the associated groups, they chose finite simple groups and designed associated puzzles. In particular, Kriz and Siegel designed puzzles for the Mathieu groups M₁₂ and M₂₄, as well as for the Conway group Co₀. Inspired by Kriz and Siegel's work, in the winter and spring of 2010 math majors Erica Chesley, Zack Starer-Stor, and Emma Zhou set out to do the same thing for other finite simple groups. They considered many puzzles and a variety of groups, and eventually designed a beautiful puzzle for the Higman-Sims group, a finite simple group of order 44,352,000 which was discovered by Higman and Sims in the late 1960s.

In 2002 Everitt, Littlejohn, and Wellman introduced the Legendre-Stirling numbers in connection with a differential operator related to Legendre polynomials. As their name suggests, the Legendre-Stirling numbers of the first and second kinds generalize the Stirling numbers of the first and second kinds, which count permutations according to length and number of cycles and set partitions according the number of elements and number of blocks, respectively. In 2008 Andrews and Littlejohn gave a combinatorial interpretation of the Legendre-Stirling numbers of the second kind in terms of a certain type of set partition, and in early 2009 I gave a combinatorial interpretation of the Legendre-Stirling numbers of the first kind in terms of pairs of permutations. In the summer of 2009 Alex Fisher used these two combinatorial interpretations to give combinatorial proofs of some identities involving Legendre-Stirling numbers, which generalize identities involving Stirling numbers. Alex also took this work a step further, proving generalizations of these identities for a wide array of number triangles whose entries satisfy a recurrence similar to that of Pascal's triangle. Alex presented his results at the annual Pi Mu Epsilon student conference at St. John's University in April of 2010. You can see a poster describing Alex's results here.