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Chapter 3: Problem 160

Explain the relationship between partitions of \(k\) into \(n\) parts and lists \(x_{1}, x_{2}, \ldots, x_{n}\) of positive integers that add to \(k\) with \(x_{1} \geq x_{2} \geq\) \(\ldots \geq x_{n} .\) Such a representation of a partition is called a decreasing list representation of the partition.

Short Answer

Expert verified
Partitions of k into n parts correspond to lists of n positive integers summing to k in decreasing order.

Step by step solution

01

Understanding Partitions of k into n parts

A partition of an integer k into n parts is a way of writing k as a sum of n positive integers. For example, if k = 7 and n = 3, one possible partition is 3 + 2 + 2 = 7.
02

Definition of Lists Adding to k

Lists such as \( x_{1}, x_{2}, \ldots, x_{n} \) are sequences of n positive integers that sum to k. For instance, for k = 7 and n = 3, a possible list is [3, 2, 2].
03

Imposing Decreasing Order

To achieve a decreasing list representation, the list must be ordered such that \( x_{1} \geq x_{2} \geq \ldots \geq x_{n} \). For example, the list [3, 2, 2] meets this criterion.
04

Establishing the Relationship

Every partition of k into n parts can be viewed as a list of n positive integers that sum to k. Each such list can be rearranged into a decreasing order, creating a unique decreasing list representation of the partition.
05

Conclusion

The relationship is that partitions of k into n parts correspond directly to lists of n positive integers summing to k, with each partition uniquely represented by a decreasing list.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integer Partitions
In combinatorics, an integer partition is a way of writing a positive integer, say k, as a sum of other positive integers. The order in which these integers are written does not matter. For example, the integer 4 can be partitioned in the following ways:
  • 4
  • 3 + 1
  • 2 + 2
  • 2 + 1 + 1
  • 1 + 1 + 1 + 1
Each such combination is called a partition of 4. When we refer to partitioning k into n parts, we want to find n positive integers that add up to k. For instance, if k = 7 and n = 3, then 7 can be partitioned into numbers like 3, 2, and 2. That way, 3 + 2 + 2 = 7.
Decreasing List Representation
A decreasing list representation is a specific way to write the partitions of an integer. Here, the integers in the list are arranged in non-increasing order. This means each part is either equal to or smaller than the previous part.
For example, let's revisit partitioning 7 into 3 parts:
  • Without any order: [2, 3, 2]
  • In decreasing order: [3, 2, 2]
To represent partitions as decreasing lists, we arrange them such that each integer is greater than or equal to the next one. This unique representation helps us easily compare and sort partitions. Mathematically, we write a decreasing list representation as follows: \( x_1 \geq x_2 \geq \ldots \geq x_n \).
Sum of Integers
The sum of integers in a partition must always equal the original integer k. This is paramount in understanding partitions.
For example, for k = 7 and n = 3, each partition must sum to 7:
  • [3, 2, 2]: 3 + 2 + 2 = 7
  • [4, 2, 1]: 4 + 2 + 1 = 7
  • [5, 1, 1]: 5 + 1 + 1 = 7
Ensuring that the sum of all parts equals k is essential to maintain the validity of the partition. The relationship between the sums and the decreasing order helps form a complete understanding of how partitions can be uniquely represented.

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Most popular questions from this chapter

A composition of the integer \(k\) into \(n\) parts is a list of \(n\) positive integers that add to \(k\). How many compositions are there of an integer \(k\) into \(n\) parts? (h)

Each function from a \(k\) -element set \(K\) to an \(n\) -element set \(N\) is a function from \(K\) onto some subset of \(N\). If \(J\) is a subset of \(N\) of size \(j\), you know how to compute the number of functions that map onto \(J\) in terms of Stirling numbers. Suppose you add the number of functions mapping onto \(J\) over all possible subsets \(J\) of \(N\). What simple value should this sum equal? Write the equation this gives you. (h)

A multiset chosen from a set \(S\) may be thought of as a subset with repeated elements allowed. For example the multiset of letters of the word Mississippi is \(\\{i, i, i, i, m, p, p, s, s, s, s\\} .\) To determine a multiset we must say how many times (including, perhaps, zero) each member of \(S\) appears in the multiset. The number of times an element appears is called its multiplicity. The size of a multiset chosen from \(S\) is the total number of times any member of \(S\) appears. For example, the size of the multiset of letters of Mississippi is \(11 .\) What is the number of multisets of size \(k\) that can be chosen from an \(n\) -element set? (h)

In how many ways may we put \(k\) identical books onto \(n\) shelves if each shelf must get at least one book?

Suppose we wish to place \(k\) distinct books onto the shelves of a bookcase with \(n\) shelves. For simplicity, assume for now that all of the books would fit on any of the shelves. Also, let's imagine pushing the books on a shelf as far to the left as we can, so that we are only thinking about how the books sit relative to each other, not about the exact places where we put the books. Since the books are distinct, we can think of a the first book, the second book and so on. (a) How many places are there where we can place the first book? (b) When we place the second book, if we decide to place it on the shelf that already has a book, does it matter if we place it to the left or right of the book that is already there? (c) How many places are there where we can place the second book? (h) (d) Once we have \(i-1\) books placed, if we want to place book \(i\) on a shelf that already has some books, is sliding it in to the left of all the books already there different from placing it to the right of all the books already or between two books already there? (e) In how many ways may we place the \(i\) th book into the bookcase? (h) (f) In how many ways may we place all the books?
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