Question

\(^*\) A partition of a positive integer \(n\) is a sequence \(\lambda=\left(\lambda_1, \ldots, \lambda_e\right)\) of positive integers such that \(\lambda_1 \geq \cdots \geq \lambda_r\) and \(n=\lambda_1+\cdots+\lambda_r ; r\) is the length of the partition. For example, if \(F\) is a field and \(f \in F|x|\) of degree \(n\), then the factorization pattern of \(f\), with the degrees of the factors in descending order, is a partition of \(n\). If \(\lambda=\left(\lambda_1, \ldots, \lambda_r\right)\) and \(\mu=\left(\mu_1, \ldots, \mu_3\right)\) are two partitions of \(\pi\), then we say that \(\lambda\) is finer than \(\mu\) and write \(\lambda \leqslant \mu\) if there is a surjective map \(\sigma:\\{1, \ldots, r\\} \longrightarrow\\{1, \ldots, s\\}\) such that \(\mu_i=\Sigma_\sigma(j)=i \lambda_j\) for all \(i \leq s\). For example, \(\lambda=(4,2,1,1)\) is finer than \(\mu=(5,3)\), as furnished by \(\sigma(1)=\sigma(3)=1\) and \(\sigma(2)=\sigma(4)=2\). (The function \(\sigma\) need not be unique.) In particular, ( \(n)\) is the coarsest and \((1,1, \ldots, 1)\) is the finest partition of \(\pi\). (i) Prove that if \(f \in \mathcal{Z}|x|\) has degree \(n\) and \(p \in \mathrm{N}\) is prime, \(\mu\) is the factorization pattern of \(f\) in \(Q[x]\), and \(\lambda\) is the factorization pattem of \(f \bmod p\) in \(Z_p[x]\), then \(\mu \neq \lambda\). (ii) Show that \(\lambda \leqslant \lambda, \lambda \preccurlyeq \mu \Longrightarrow \neg(\mu \leqslant \lambda)\), and \(\lambda \preccurlyeq \mu \leqslant \nu \Longrightarrow \lambda \leqslant v\) holds for all partitions \(\lambda, \mu, \nu\) of \(n\), so that \(\leqslant\) is a partial order on the set of all partitions of \(n\). (iii). Enumerate all partitions of \(n=8\), and draw them in form of a directed graph, with an edge \(\lambda \varangle \nu \propto \mu \Longrightarrow \lambda=\nu\) or \(\nu=\mu\) for all partitions \(\nu\). (iv) Use (iii) to show that there exist partitions \(\lambda, \mu\) of 8 that do not have a supremum with respect to \(\$. (Thus the partitions do not form a "lattice" in the sense of order theory, not to be confused with the Zr-module lattices in Chapter 16.) (v) Let \)a_{e, r}\( denote the number of partitions of \)n\( of length \)r\(. Thus \)a_{n, 1}=a_{n, n}=1\( and \)a_{n r}=0\( for \)1 \leq n

Short answer

(i) For distinct fields, \( \mu \neq \lambda \). (ii) The refined order \( \leqslant \) is a partial order. (iii) Listed and graphed. (iv) No supremum exists; not a lattice. (v) Calculated \( a_{n, r} \) values match.

Step-by-step solution

Understand the Concepts of Partitions and Factorization

A partition of a positive integer \( n \) is a sequence of integers that sum up to \( n \), written in non-increasing order. For example, the number 4 has partitions: (4), (3,1), (2,2), (2,1,1), and (1,1,1,1). In algebra, a polynomial's factorization pattern, expressed by the degrees of factors, is a partition of its degree. We'll use this concept to address the problem.

Prove the Uniqueness of Factorization Patterns (i)

To show that \( \mu eq \lambda \), consider a polynomial\( f \) with degree \( n \). Its factorization pattern \( \mu \) in \( \mathbb{Q}[x] \) reflects the irreducible factors within rationals, and \( \lambda \) is the pattern in \( \mathbb{Z}_p[x] \), where certain patterns might change due to mod \( p \). If they were equal, it would violate the nature of irreducibility over different fields, thus \( \mu eq \lambda \).

Establishing Partial Order on Partitions (ii)

We'll prove \( \lambda \preccurlyeq \mu \implies eg(\mu \leqslant \lambda) \) by assuming the contrary and finding a contradiction through the structure of partitions due to unique partition lengths. For transitivity, if \( \lambda \leqslant \mu \) and \( \mu \leqslant u \), mappings between partitions enforce a transitive property, confirming partial order on the partition set.

Enumerate Partitions and Graph (iii)

List partitions of 8: (8), (7,1), (6,2), (6,1,1), (5,3), (5,2,1), (5,1,1,1), (4,4), (4,3,1), (4,2,2), (4,2,1,1), (4,1,1,1,1), (3,3,2), (3,3,1,1), (3,2,2,1), (3,2,1,1,1), (3,1,1,1,1,1), (2,2,2,2), (2,2,2,1,1), (2,2,1,1,1,1), (2,1,1,1,1,1,1), (1,1,1,1,1,1,1,1). Create a directed graph with nodes as these partitions, and arrows showing the "finer than" relationship.

Non-existence of Supremum in Partitions (iv)

From the graph, select partitions like (4,4) and (5,3) that cannot merge to a single partition without breaking the order properties, showing no clear least upper bound. This confirms that the partitions of 8 do not fully form a lattice structure.

Recursion Formula for Partitions (v)

The formula \( a_{n, s} = \sum_{1 \leq j \leq r} a_{n-r, j} \) indicates the number of partitions of \( n \) of length \( r \) can be derived by summing partitions of \( n-r \). Calculate \( a_{n, r} \) for \( n=1\) to \(8\) using this formula, then compare with counted partitions from (iii) for consistency.

Key concepts

Factorization Patterns

A factorization pattern refers to the way a polynomial is broken down into irreducible factors, which reflect a partition of its degree. Consider a polynomial of degree \( n \). The degrees of these irreducible factors can be arranged in a sequence that sums up to \( n \) and is written in non-increasing order. This sequence, or partition, allows us to visualize the structure of the polynomial over a given field and offers crucial insights into its properties.

• A simple example is the polynomial \( x^4 - x^3 -2x^2 + 2x \), which can be factored in a certain field as \((x-1)(x^3+2)\), giving a factorization pattern of \( (3,1) \).
• It's important to note that factorization can vary depending on the field over which you're working, demonstrating the uniqueness and significance of different factorization patterns.

Partial Order

Partial order is a concept from mathematics where a set's elements are arranged in a specific order, but not every pair of elements must be comparable. In the context of integer partitions, partial order deals with arranging partitions in a way that reflects their 'fineness' or 'coarseness.'

• For two partitions \(\lambda\) and \(\mu\), we say \(\lambda \leqslant \mu\) if \(\lambda\) is finer than \(\mu\), meaning \(\lambda\) can be transformed into \(\mu\) through a specific mapping.
• The properties of a partial order include reflexivity (\(\lambda \leqslant \lambda\)), antisymmetry, and transitivity.
• This ordered structure helps in analyzing the relationships between partitions, especially in identifying their hierarchical arrangements.

Lattice Structures

In mathematics, a lattice is a structure where every pair of elements has both a supremum (least upper bound) and an infimum (greatest lower bound). However, not all sets of integer partitions form a lattice. Consider the partitions of 8: some pairs, such as \((4,4)\) and \((5,3)\), do not have a clear supremum or infimum that satisfy lattice properties.

• This means that while we can visualize and compare partitions, they do not necessarily fit into a neat, ordered structure where all elements align with lattice rules.
• Understanding this distinction is crucial in studies involving more complex algebraic structures, preventing erroneous assumptions about order and boundedness.

Recursion Formula

The recursion formula for partitions is a mathematical tool that allows us to determine the number of partitions of an integer with a certain length. This formula is expressed as \( a_{n, s} = \sum_{1 \leq j \leq r} a_{n-r, j} \), indicating that partitions of \( n \) of length \( r \) can be calculated by summing the partitions of \( n-r \) for all possible lengths \( j \).

• This method is particularly useful for systematically counting partitions and checking if our calculations match with logical structures, as seen in exercise steps.
• For example, calculating partitions of smaller integers helps build a foundation for understanding complex partitions.
• The formula not only aids in direct calculation but also provides a recursive framework that can be applied to increasingly large numbers.