Prime Power Divisors of Multinomial and q-Multinomial Coefficients

On Common Divisors of Multinomial Coefficients

Erdős and Szekeres showed in 1978 that for any four positive integers satisfying m1 + m2 = n1 +n2, the two binomial coefficients (m1 +m2)!/m1!m2! and (n1 +n2)!/n1!n2! have a common divisor > 1. The analogous statement for families of k k-nomial coefficients (k > 1) was conjectured in 1997 by David Wasserman. Erdős and Szekeres remark that if m1, m2, n1, n2 as above are all > 1, there is probabl...

The Andrews-gordon Identities and Q-multinomial Coefficients

We prove polynomial boson-fermion identities for the generating function of the number of partitions of n of the form n = ∑L−1 j=1 jfj, with f1 ≤ i−1, fL−1 ≤ i ′−1 and fj+fj+1 ≤ k. The bosonic side of the identities involves q-deformations of the coefficients of xa in the expansion of (1 + x + · · · + xk)L. A combinatorial interpretation for these q-multinomial coefficients is given using Durfe...

The Number of Multinomial Coefficients

Fermat Numbers in Multinomial Coefficients

In 2001 Luca proved that no Fermat number can be a nontrivial binomial coefficient. We extend this result to multinomial coefficients.

On the Number of Distinct Multinomial Coefficients

We study M(n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where pP(n) is the number of partitions of n into primes and p(n) is the total number of partitions of n. To use methods from commutative algebra, we encode partitions and multinomial...