Lemma 3.4 in this interesting paper by A. Kleshchev and D. Nash requires a combinatorial result that I hadn’t seen before. Here is a version of it, converted into a typical mathematical game (i.e. only entertaining for people who have nothing better to do).
We work in the ring where is an indeterminate. Imagine a row of coins, numbered from at the far left to at the far right. Your task is to remove all coins. As you do this, you must keep track of an element . At the start of the game . If, after taking coin , there are coins to its left, and empty gaps, then the rules of the game require you to multiply by .
For example, if and we take coins in the order , then at the end of the game .
Now let be the sum of the final outcomes over all orders in which we might remove the coins. The remarkable result is is the quantum generalization of the factorial function, i.e.
where the quantum number is defined by
It is often useful to write in the alternative form
In this form the important property that as is obvious. It follows that . Correspondingly, if we play the game with , then the final outcome is always , so this limit recovers the obvious fact that there are different ways in which we may remove the coins.
The result that is easily proved by induction on . If there is just one coin then the final outcome is always . Now suppose the result holds for coins, and suppose that we begin an -coin game by removing coin . This gives us an immediate contribution of . Imagine that we ignore the gap at coin and continue playing the game as if we had started with coins. By induction, the sum of over all possible continuations is . This has to be scaled by to take into account the gap we ignored when removing coins . (The order in which we remove these coins is irrelevant.) Hence
Since , it follows that