Question

Find the Cantor digits \({{\rm{a}}_{\rm{1}}}{\rm{,}}{{\rm{a}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{a}}_{{\rm{n - 1}}}}\) that correspond to these permutations.

a) 246531

b) 12345

c) 654321

Short answer

1. The cantor digits\({{\rm{a}}_{\rm{1}}}{\rm{,}}{{\rm{a}}_{\rm{2}}}{\rm{,}}{{\rm{a}}_{\rm{3}}}{\rm{,}}{{\rm{a}}_{\rm{4}}}{\rm{,}}{{\rm{a}}_{\rm{5}}}\)that correspond to 246531 are\({\rm{1,1,2,2,3}}\).

b. The cantor digits \({{\rm{a}}_{\rm{1}}}{\rm{,}}{{\rm{a}}_{\rm{2}}}{\rm{,}}{{\rm{a}}_{\rm{3}}}{\rm{,}}{{\rm{a}}_{\rm{4}}}\) that correspond to 12345 are\({\rm{0,0,0 \& 0}}\).

c. The cantor digits that correspond to 654321 are:

\(\begin{array}{l}{{\rm{a}}_{\rm{1}}}{\rm{ = 1}}\quad \\{{\rm{a}}_{\rm{4}}}{\rm{ = 4}}\\{{\rm{a}}_{\rm{2}}}{\rm{ = 2}}\quad \\{{\rm{a}}_{\rm{5}}}{\rm{ = 5}}\\{{\rm{a}}_{\rm{3}}}{\rm{ = 3}}\end{array}\)

Step-by-step solution

Definition of Concept

Functions: It is a expression, rule or law which defines a relationship between one variable and another variables.

Find the Cantor digits

(a)

Considering the given information:

The given permutation is 246531.

Using the following concept:

Cantor expansion of x consists of coefficients \({{\rm{a}}_{\rm{n}}}{\rm{, \ldots \ldots }}{\rm{.}}{{\rm{a}}_{\rm{1}}}\) such that.

\(0 \le {a_i} \le i{\rm{ And }}x = {a_n}(n!) + \ldots \ldots \ldots .. + {a_2}(2!) + {a_1}(1!)\)

\({{\rm{a}}_{\rm{1}}}{\rm{ = }}\)The number of integers less than 2 that follow \({\rm{2 = 1}}\)

\({{\rm{a}}_{\rm{2}}}{\rm{ = }}\)The number of integers less than 3 that follows \({\rm{3 = 1}}\)

\({{\rm{a}}_{\rm{3}}}{\rm{ = }}\)The number of integer less than 4 that follows \({\rm{4 = 2}}\)

\({{\rm{a}}_{\rm{4}}}{\rm{ = }}\)The number of integer less than 5 that follows \({\rm{5 = 2}}\)

\({{\rm{a}}_{\rm{5}}}{\rm{ = }}\)The number of integers less than 6 that follows \({\rm{6 = 3}}\)

So, the\({{\rm{a}}_{\rm{1}}}{\rm{,}}{{\rm{a}}_{\rm{2}}}{\rm{,}}{{\rm{a}}_{\rm{3}}}{\rm{,}}{{\rm{a}}_{\rm{4}}}{\rm{,}}{{\rm{a}}_{{\rm{6 - 1}}}}{\rm{ = 1,1,2,2,3}}\).

In addition, the Cantor equation is

\(\begin{array}{l}{\rm{ = }}{{\rm{a}}_{\rm{1}}}{\rm{(1!) + }}{{\rm{a}}_{\rm{2}}}{\rm{(2!) + }}{{\rm{a}}_{\rm{3}}}{\rm{(3!) + }}{{\rm{a}}_{\rm{4}}}{\rm{(4!) + }}{{\rm{a}}_{\rm{5}}}{\rm{(5!)}}\\{\rm{ = 1(1) + 1(2!) + 2(3!) + 2(4!) + (5!)}}\\{\rm{ = 12 + 12 + 48 + 360}}\\{\rm{ = 423}}\end{array}\)

Therefore, the cantor digits\({{\rm{a}}_{\rm{1}}}{\rm{,}}{{\rm{a}}_{\rm{2}}}{\rm{,}}{{\rm{a}}_{\rm{3}}}{\rm{,}}{{\rm{a}}_{\rm{4}}}{\rm{,}}{{\rm{a}}_{\rm{5}}}\)that correspond to 246531 are\({\rm{1,1,2,2,3}}\).

Find the Cantor digits

(b)

Considering the given information:

The given permutation is 12345.

Using the following concept:

Cantor expansion of x consists of coefficients \({{\rm{a}}_{\rm{n}}}{\rm{, \ldots \ldots }}{\rm{.}}{{\rm{a}}_{\rm{1}}}\) such that.

\(0 \le {a_i} \le i{\rm{ And }}x = {a_n}(n!) + \ldots \ldots \ldots .. + {a_2}(2!) + {a_1}(1!)\)

\({{\rm{a}}_{\rm{1}}}{\rm{ = }}\)The number of integers less than 2 that follow \({\rm{2 = 0}}\)

\({{\rm{a}}_{\rm{2}}}{\rm{ = }}\)The number of integers less than 3 that follows \({\rm{3 = 0}}\)

\({{\rm{a}}_{\rm{3}}}{\rm{ = }}\)The number of integer less than 4 that follows \({\rm{4 = 0}}\)

\({{\rm{a}}_{\rm{4}}}{\rm{ = }}\)The number of integer less than 5 that follows \({\rm{5 = 0}}\)

So, the cantor digits are\({\rm{0,0,0\& 0}}\).

In addition, the Cantor equation is

\(\begin{array}{l}{\rm{ = }}{{\rm{a}}_{\rm{1}}}{\rm{(1!) + }}{{\rm{a}}_{\rm{2}}}{\rm{(2!) + }}{{\rm{a}}_{\rm{3}}}{\rm{(3!) + }}{{\rm{a}}_{\rm{4}}}{\rm{(4!)}}\\{\rm{ = 0}}\end{array}\)

Therefore, the cantor digits \({{\rm{a}}_{\rm{1}}}{\rm{,}}{{\rm{a}}_{\rm{2}}}{\rm{,}}{{\rm{a}}_{\rm{3}}}{\rm{,}}{{\rm{a}}_{\rm{4}}}\) that correspond to 12345 are\({\rm{0,0,0 \& 0}}\).

Find the Cantor digits

(c)

Considering the given information:

The given permutation is 654321.

Using the following concept:

Cantor expansion of x consists of coefficients \({{\rm{a}}_{\rm{n}}}{\rm{, \ldots \ldots }}{\rm{.}}{{\rm{a}}_{\rm{1}}}\) such that.

\(0 \le {a_i} \le i{\rm{ And }}x = {a_n}(n!) + \ldots \ldots \ldots .. + {a_2}(2!) + {a_1}(1!)\)

\({{\rm{a}}_{\rm{1}}}{\rm{ = }}\)The number of integers less than 2 that follow \({\rm{2 = 1}}\)

\({{\rm{a}}_{\rm{2}}}{\rm{ = }}\)The number of integers less than 3 that follows \({\rm{3 = 2}}\)

\({{\rm{a}}_{\rm{3}}}{\rm{ = }}\)The number of integer less than 4 that follows \({\rm{4 = 3}}\)

\({{\rm{a}}_{\rm{4}}}{\rm{ = }}\)The number of integer less than 5 that follows \({\rm{5 = 4}}\)

\({{\rm{a}}_{\rm{5}}}{\rm{ = }}\)The number of integers less than 6 that follows \({\rm{6 = 5}}\)

So, the cantor digits are\({\rm{1,2,3,4,5}}\).

In addition, the Cantor equation is

\(\begin{array}{l}{\rm{ = }}{{\rm{a}}_{\rm{1}}}{\rm{(1!) + }}{{\rm{a}}_{\rm{2}}}{\rm{(2!) + }}{{\rm{a}}_{\rm{3}}}{\rm{(3!) + }}{{\rm{a}}_{\rm{4}}}{\rm{(4!) + }}{{\rm{a}}_{\rm{5}}}{\rm{(5!)}}\\{\rm{ = 1(1) + 2(2) + 3(6) + 4(24) + 5(120)}}\\{\rm{ = 719}}\end{array}\)

Therefore, the required cantor digits are:

\(\begin{array}{l}{{\rm{a}}_{\rm{1}}}{\rm{ = 1}}\quad \\{{\rm{a}}_{\rm{4}}}{\rm{ = 4}}\\{{\rm{a}}_{\rm{2}}}{\rm{ = 2}}\quad \\{{\rm{a}}_{\rm{5}}}{\rm{ = 5}}\\{{\rm{a}}_{\rm{3}}}{\rm{ = 3}}\end{array}\)