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Chapter 5: Problem 14

Write a computer program (or develop an algorithm) to compute the Stirling numbers \(S(m, n)\) when \(1 \leq m \leq 12\) and \(1 \leq n \leq m\).

Short Answer

Expert verified
The Stirling Number can be quickly computed for any \(m, n \leq 12\) using an iterative approach backed by dynamic programming. The core idea is to initialize a 13 x 13 matrix, fill it in using the recurrence relation for Stirling Numbers, and then simply read off the required values from the computed matrix.

Step by step solution

01

Initialize the matrix

Firstly, considering the values of m and n ranging from 1 to 12, create a matrix \(S[m][n]\) of dimension 13 x 13. Make sure to initialize it to zero for all values of \(m > 0, n = 0\) and \(m > 0, n > m\), and set \(S[m][0] = 1\) for \(m = 0, n = 1\).
02

Fill in the matrix using the recurrence relation

Begin iterating over the matrix, starting from \(m = n = 1\), and for each cell (m, n), compute the value as \(S[m][n] = n * S[m - 1][n] + S[m - 1][n - 1]\). Proceed in this manner till the matrix is completely filled.
03

Obtain the result

After the matrix is completely filled, the Stirling Number \(S(m, n)\) will be the value of the cell in the m-th row and n-th column of the matrix. This matrix can now be used to quickly lookup Stirling numbers for given m and n within the range.

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Key Concepts

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

Combinatorial Mathematics
Combinatorial mathematics is a branch of mathematics that deals with counting, arrangement, and combination of sets of elements. In essence, it involves the study of finite or discrete structures. This field includes a variety of topics such as graph theory, enumeration, and partitioning, but it's particularly relevant to our discussion due to its involvement with Stirling numbers. Stirling numbers themselves are used to solve problems related to permutations and combinations where certain constraints are imposed. They come in two types, which are the Stirling numbers of the first kind and the Stirling numbers of the second kind, each providing different combinatorial insights.
Computing Stirling Numbers
Computing Stirling numbers, specifically of the second kind, is about understanding the ways to partition a set of objects into non-empty subsets. The Stirling number of the second kind, denoted as \(S(m, n)\), answers the question: 'In how many ways can m objects be partitioned into n non-empty subsets?' These numbers play a crucial role in various combinatorial problems and are particularly notable for their appearance in many mathematical formulas involving permutations and partitions as well as their relationships with Bell numbers, which count the number of possible partitions of a set.
Recurrence Relation
A recurrence relation is a sequence of numbers in which each term after the first is determined by the terms that preceded it, in combination with certain fixed rules. For Stirling numbers of the second kind, the recurrence relation is given by \(S(m, n) = n * S(m - 1, n) + S(m - 1, n - 1)\), where the base cases are defined as \(S(0, 0) = 1\), \(S(m, 0) = 0\) for \(m > 0\), and \(S(m, n) = 0\) for \(n > m\) or \(n = 0\). This recurrence relation allows us to build up the computation of Stirling numbers in a systematic way using previously computed values, which is a powerful technique in both mathematical proof and algorithm design.
Algorithm Development
The development of an algorithm to compute Stirling numbers first involves understanding and implementing the underlying recurrence relation. Then, it requires translating this understanding into a series of structured steps programmed in a computer.

Initial Setup

  • Create a two-dimensional array to represent the matrix.
  • Initialize the matrix following the base cases of the recurrence relation.

Iterative Computation

  • Fill in the matrix by iteratively applying the recurrence relation to compute new values based on previously computed ones.
  • Optimize the process to avoid unnecessary computations and ensure efficient use of memory and compute resources.
By following these principles, the algorithm you develop can efficiently compute Stirling numbers for any given pair of integers within the pre-set bounds.

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Most popular questions from this chapter

a) In how many ways can \(31,100,905\) be factored into three factors, each greater than 1 , if the order of the factors is not relevant? b) Answer part (a), assuming the order of the three factors is relevant. c) In how many ways can one factor \(31,100,905\) into two or more factors where each factor is greater than 1 and no regard is paid to the order of the factors? d) Answer part (c), assuming the order of the factors is to be taken into consideration.

Verify that \(\sum_{k=0}^{n}(-1)^{k}\left({ }_{n-k}{n}\right)(n-k)^{m}=0\) for \(n=5\) and \(m=2,3,4\).

Let \(A, B\) be any sets. a) Prove that I) \((A \times B) \cap(B \times A)=(A \cap B) \times(A \cap B)\); and, ii) \((A \times B) \cup(B \times A) \subseteq(A \cup B) \times(A \cup B)\). b) Provide an example to show that \((A \cup B) \times\) \((A \cup B)\) need not be a subset of \((A \times B) \cup\) \((B \times A)\).

Let \(f, g: \mathbf{Z}^{+} \rightarrow \mathbf{R}\) where \(f(n)=n\) and \(g(n)=\log _{2} n\), for \(n \in \mathbf{Z}^{+}\). Show that \(g \in O(f)\) but \(f \notin O(\mathrm{~g}) .\left(\right.\) Hint \(\lim _{n \rightarrow \infty} \frac{n}{\log _{2} n}=+\infty .\) This requires the use of calculus. \()\)

Let \(A\) be a set with \(|A|=n\). a) How many closed binary operations are there on \(A\) ? b) A closed ternary (3-ary) operation on \(A\) is a function \(f: A \times A \times A \rightarrow A\). How many closed ternary operations are there on \(A\) ? c) A closed \(k\)-ary operation on \(A\) is a function \(f: A_{1} \times A_{2} \times \cdots \times A_{k} \rightarrow A\), where \(A_{i}=A\), for all \(1 \leq i \leq k\). How many closed \(k\)-ary operations are there on \(A\) ? d) A closed \(k\)-ary operation for \(A\) is called commutative if $$ f\left(a_{1}, a_{2}, \ldots, a_{k}\right)=f\left(\pi\left(a_{1}\right), \pi\left(a_{2}\right), \ldots, \pi\left(a_{k}\right)\right), $$ where \(a_{1}, a_{2}, \ldots, a_{k} \in A\) (repetitions allowed), and \(\pi\left(a_{1}\right), \pi\left(a_{2}\right), \ldots, \pi\left(a_{k}\right)\) is any rearrangement of \(a_{1}, a_{2}, \ldots, a_{k}\). How many of the closed \(k\)-ary operations on \(A\) are commutative?
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