Presentation Title

On the Behavior of Integer Partitions and Carry Sequences

Faculty Mentor

Mark Kozek

Start Date

18-11-2017 11:00 AM

End Date

18-11-2017 11:15 AM

Location

9-285

Session

Physical Sciences 2

Type of Presentation

Oral Talk

Subject Area

physical_mathematical_sciences

Abstract

The integer partitions of a positive integer $n$ are all the ways that $n$ can be written as a sum of positive integers. For example, $4+8$ and $3+4+5$ are partitions of 12. Carry sequences keeps track of the place values in a number's decimal expansion where we had to ``carry'' a digit to while adding. Computing the sum $4+8$ in base-10, we find the carry sequence is $(1,0)$, because when we add 8 and 4 in the units place, we end up having to ``carry'' 1 to the tens place. However, we can do this operation in any base-$b$. In fact, there are times where the digits of the carries sequence match the digits of the base-$b$ representation of $n$. For example, the partition $4+8$ in base-3, $(0,1,1)+(0,2,2)$, gives the carry sequence of $(1,1,0)$ when we add them together, while the base-3 representation of their sum 12 is $(1,1,0)_3$. We define this as the equal carries property, that is to say, when the carry sequence of a partition is equal to its sum in a given base-$b$. In this talk we prove a criterion for determining whether a partition has the equal carries property and provide observations pertaining to such partitions.

Summary of research results to be presented

We will first be introducing integer partitions and carry sequences. We continue on to define an equal carries partition and define p-part hyper b-ary representations of integers. Using these representations, we prove a criterion to determine whether a partition has the equal carries property. Using this criterion, we are able to observe many interesting behaviors and patterns that arise amongst partitions. We discuss all of these patterns in-depth as well as provide detailed examples that sketch out potential proofs.

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Nov 18th, 11:00 AM Nov 18th, 11:15 AM

On the Behavior of Integer Partitions and Carry Sequences

9-285

The integer partitions of a positive integer $n$ are all the ways that $n$ can be written as a sum of positive integers. For example, $4+8$ and $3+4+5$ are partitions of 12. Carry sequences keeps track of the place values in a number's decimal expansion where we had to ``carry'' a digit to while adding. Computing the sum $4+8$ in base-10, we find the carry sequence is $(1,0)$, because when we add 8 and 4 in the units place, we end up having to ``carry'' 1 to the tens place. However, we can do this operation in any base-$b$. In fact, there are times where the digits of the carries sequence match the digits of the base-$b$ representation of $n$. For example, the partition $4+8$ in base-3, $(0,1,1)+(0,2,2)$, gives the carry sequence of $(1,1,0)$ when we add them together, while the base-3 representation of their sum 12 is $(1,1,0)_3$. We define this as the equal carries property, that is to say, when the carry sequence of a partition is equal to its sum in a given base-$b$. In this talk we prove a criterion for determining whether a partition has the equal carries property and provide observations pertaining to such partitions.