428 research outputs found
Higher Order SPT-Functions
Andrews' spt-function can be written as the difference between the second
symmetrized crank and rank moment functions. Using the machinery of Bailey
pairs a combinatorial interpretation is given for the difference between higher
order symmetrized crank and rank moment functions. This implies an inequality
between crank and rank moments that was only know previously for sufficiently
large n and fixed order. This combinatorial interpretation is in terms of a
weighted sum of partitions. A number of congruences for higher order
spt-functions are derived.Comment: 21 pages (previous version was 19 pages), added reference to Andrews
and Rose's recent paper, MacMahon's paper and OEIS, changed some wordin
The Arithmetic of Multiple Harmonic Sums.
This dissertation concerns the arithmetic of a family of rational numbers called multiple harmonic sums. These sums are finite truncations of multiple zeta values. We consider multiple harmonic sums whose truncation point is one less than a prime.
We derive families of congruences, involving multiple harmonic sums, for binomial coefficients and for values of the Kubota-Leopoldt p-adic L-function at positive integers. Congruences in our families hold modulo arbitrarily large powers of prime. We also set up a framework for studying congruences among multiple harmonic sums, which is related to a framework used in the study of multiple zeta values.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/99893/1/rosenjh_1.pd
S-Restricted Compositions Revisited
An S-restricted composition of a positive integer n is an ordered partition
of n where each summand is drawn from a given subset S of positive integers.
There are various problems regarding such compositions which have received
attention in recent years. This paper is an attempt at finding a closed- form
formula for the number of S-restricted compositions of n. To do so, we reduce
the problem to finding solutions to corresponding so-called interpreters which
are linear homogeneous recurrence relations with constant coefficients. Then,
we reduce interpreters to Diophantine equations. Such equations are not in
general solvable. Thus, we restrict our attention to those S-restricted
composition problems whose interpreters have a small number of coefficients,
thereby leading to solvable Diophantine equations. The formalism developed is
then used to study the integer sequences related to some well-known cases of
the S-restricted composition problem
Counting With Irrational Tiles
We introduce and study the number of tilings of unit height rectangles with
irrational tiles. We prove that the class of sequences of these numbers
coincides with the class of diagonals of N-rational generating functions and a
class of certain binomial multisums. We then give asymptotic applications and
establish connections to hypergeometric functions and Catalan numbers
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