Nested Sequence of Vector Spaces and Flags

In linear algebra, a flag is a strictly nesting sequence of subspaces of a finite dimensional vector space  
\[\mathbf{0} \subset V_1 \subset V_2 \subset ... \subset V_n =V\]

If the dimension of  
\[d_0 \lt d_1 \lt d_2 \lt d_3 \lt ...\lt d_k =n\]

  is the dimension of  
. A flag is called a complete flag if  
\[d_i =i\]
, otherwise it is called a partial flag. The sequence  
\[d_1, \: d_2 , \: d_3, \: d_4 \: ... \: d_k\]
  is called the signature of the flag.