Room PK-4323, Pav. André-Aisenstadt, 201 Ave. President-Kennedy, H2X 2Y7, CA
Enumeration of binomial coefficients by their p-adic valuations
In 1947 Nathan Fine obtained a beautiful formula for the number of binomial coefficients binomial(n, m), for fixed n, that are not divisible by p: Write n in base p, add 1 to each digit, and multiply them all together. Subsequently, many authors found formulas counting binomial coefficients with p-adic valuation alpha for particular values of p and alpha, but a general formula remained elusive. We give a matrix product, generalizing Fine's result, for the generating function counting binomial coefficients by their p-adic valuations. A further generalization counts multinomial coefficients by their p-adic valuations.