3,736 research outputs found
MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms
We investigate a relationship between MacMahon's generalized sum-of-divisors
functions and Chebyshev polynomials of the first kind. This determines a
recurrence relation to compute these functions, as well as proving a conjecture
of MacMahon about their general form by relating them to quasi-modular forms.
These functions arise as solutions to a curve-counting problem on Abelian
surfaces.Comment: 6 Page
A new four parameter q-series identity and its partition implications
We prove a new four parameter q-hypergeometric series identity from which the
three parameter key identity for the Goellnitz theorem due to Alladi, Andrews,
and Gordon, follows as a special case by setting one of the parameters equal to
0. The new identity is equivalent to a four parameter partition theorem which
extends the deep theorem of Goellnitz and thereby settles a problem raised by
Andrews thirty years ago. Some consequences including a quadruple product
extension of Jacobi's triple product identity, and prospects of future research
are briefly discussed.Comment: 25 pages, in Sec. 3 Table 1 is added, discussion is added at the end
of Sec. 5, minor stylistic changes, typos eliminated. To appear in
Inventiones Mathematica
Partitions with fixed differences between largest and smallest parts
We study the number of partitions of with difference between
largest and smallest parts. Our main result is an explicit formula for the
generating function . Somewhat
surprisingly, is a rational function for ; equivalently,
is a quasipolynomial in for fixed . Our result generalizes to
partitions with an arbitrary number of specified distances.Comment: 5 page
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 coefficients as monomials.Comment: 16 pages, to be published in the Journal of Number Theor
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