I gave a talk at the Mathematics Teachers’ Circle of Austin on the distribution of carries in adding numbers in a given base. (The full title “Dizzying the memory of arithmetic” is supposed to be a play of words on a phrase in Macbeth…)
The question is this. Take x_1,…,x_k some n digit numbers, say in base 10. When you add them with the usual elementary school algorithm some of the columns of k digits will contribute a carry, a number in the range 0 to k-1, to the next column. What is the distribution of these carries for random x_i for large n?
Surprisingly, Diaconis and Fulman found a connection between this question and card shuffling, Foulkes characters, symmetric functions,…
I love it when we discover something new and interesting in things right under our noses.