How to resolve the algorithm Stirling numbers of the first kind step by step in the Mathematica / Wolfram Language programming language

Published on 22 June 2024 08:30 PM
#Wolfram

How to resolve the algorithm Stirling numbers of the first kind step by step in the Mathematica / Wolfram Language programming language

Table of Contents

Problem Statement

Stirling numbers of the first kind, or Stirling cycle numbers, count permutations according to their number of cycles (counting fixed points as cycles of length one). They may be defined directly to be the number of permutations of n elements with k disjoint cycles. Stirling numbers of the first kind express coefficients of polynomial expansions of falling or rising factorials. Depending on the application, Stirling numbers of the first kind may be "signed" or "unsigned". Signed Stirling numbers of the first kind arise when the polynomial expansion is expressed in terms of falling factorials; unsigned when expressed in terms of rising factorials. The only substantial difference is that, for signed Stirling numbers of the first kind, values of S1(n, k) are negative when n + k is odd. Stirling numbers of the first kind follow the simple identities:

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Stirling numbers of the first kind step by step in the Mathematica / Wolfram Language programming language

Explanation:

Line 1:

  • TableForm[Array[StirlingS1, {n = 12, k = 12} + 1, {0, 0}], TableHeadings -> {"n=" <> ToString[#] & /@ Range[0, n], "k=" <> ToString[#] & /@ Range[0, k]}]

This line creates a table form with:

* 13 rows (n = 0 to 12) * 13 columns (k = 0 to 12) * Each cell contains the value of StirlingS1[n, k] * The first row and first column are labeled with "n=" and "k=", respectively, with the corresponding values

The Array function generates a 2D array with the specified dimensions and element generation function (StirlingS1[n, k]), and the TableHeadings option adds the table headings.

Line 2:

  • Max[Abs[StirlingS1[100, #]] & /@ Range[0, 100]]

This line computes and prints the maximum absolute value of StirlingS1[100, k] for k in the range [0, 100]. It uses the Max function to find the maximum value of the list of absolute values of the StirlingS1 function evaluated at n = 100 and k from 0 to 100.

Overall Functionality:

This code generates a table that shows the values of the Stirling numbers of the first kind (StirlingS1) for n and k from 0 to 12. It also computes the maximum absolute value of StirlingS1[100, k] for k in the range [0, 100].

Source code in the wolfram programming language

TableForm[Array[StirlingS1, {n = 12, k = 12} + 1, {0, 0}], TableHeadings -> {"n=" <> ToString[#] & /@ Range[0, n], "k=" <> ToString[#] & /@ Range[0, k]}]
Max[Abs[StirlingS1[100, #]] & /@ Range[0, 100]]

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