Young Diagrams and The Ski Lift Problem
Somewhere years ago when I was in college, I came across what seemed like a simple combinatorics problem: "There are 3 ski lifts and 3 students who wish to ride them. Given that each lift has an unlimited amount of room to seat students, what is the total number of permutations of students on lifts?" As a fun bonus, rather than simply solving the problem (which is really a trivial calculation on its own) I thought it would be fun to find a generating function for any number of lifts and students. Not knowing much beyond rudimentary discrete math or combinatorics, I began to work. This at first seems like a straight-forward permutation problem: there are both students and lifts being permuted, but ultimately this is nothing a first year student couldn't reckon with. To create a generating function, it's necessary to calculate the permutations of lifts used, then multiply this by the number of permutations of students. For the lifts, there are 3 of what I’ll