Sequence

A sequence is a fundamental concept in formal order analysis that is defined as a finite, enumerated collection of objects in which repetitions are allowed and order matters. This is the basic object for analyzing patterns, relationships, and structural properties in ordered data.

Mathematical Definition

Let \(X\) is Carrier set

A sequence \(S\) is a n-tuple defined as

\[S = <s_1, s_2, ..., s_n>,\]

\[\forall i \in \{1, ..., n\} \exists s_i \in X\]

where:

• \(s_i\)​ is called the \(i\)-th element (or coordinate) of the sequence.
• \(n := |S|\) is length, \(n \in N\)

The sequence \(S\) can be also defined as a function

\[S : \{1, ..., n\} \longrightarrow X,\]

\[S(i)=s_i | i \in \{1, ..., n\}\]

Examples

Binary Sequence

A binary sequence 0110100110

represented as

\(X = \{0,1\}\)

\(S = <0,1,1,0,1,0,0,1,1,0>\)

Musical Chorus Sequence

A musical chorus for Jingle bell rock

D                Dmaj7        D6
Jingle-bell, Jingle-bell, Jingle-bell Rock.
  D                D#dim
Jingle-bell swing and
 Em           A7     Em               A7            Em A7
Jingle-bell ring. Snowin' and blowin' up bushels of fun.
Em  A9                  A7
Now the jingle-hop has begun.

\(X = \{A7, A9, D, D6, Dmaj7, D\#dim, Em\}\)

\(S = <D,Dmaj7,D6,D,D\#dim,Em,A7,Em,A7,Em,A7,Em,A9,A7>\)

DNA Sequence

A DNA sequence ATGCTAGCATGCTAGCATGCTAGC

\(X = \{A,C,T,G\}\)

\(S = <A,T,G,C,T,A,G,C,A,T,G,C,T,A,G,C,A,T,G,C,T,A,G,C>\)

English Text Sequence as char sequence

An English text sentence the quick brown fox jumps over the lazy dog

\(X = \{\ ,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}\)

\(S = <t,h,e,\ ,q,u,i,c,k,\ ,b,r,o,w,n,\ ,f,o,x,\ ,j,u,m,p,s,\ ,o,v,e,r,\ ,t,h,e,\ ,l,a,z,y,\ ,d,o,g>\)

English Text Sequence as word sequence

An English text sentence the quick brown fox jumps over the lazy dog

\(X = \{\ ,quick, fox, brown, the, over, dog, fox, lazy\}\)

\(S = <the,\ ,quick,\ ,brown,\ ,fox,\ ,jumps,\ ,over,\ ,the,\ ,lazy,\ ,dog>\)