#### 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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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.