In the analysis of this insertion sort algorithm, what does the summation mean?

Question

For this analysis of Insertion Sort, as shown in Introduction to Algorithms:

What does the summation at line 5 indicate? I'm very confused what tj is supposed to mean. Why does it not just show that it occurs n*n times or something?

Could someone clarify what it is saying?

Answer 1

tj is the number of times the while loop is executed (for the given value of j)

this is a variable that depends on the initial order of the array

Answer 2

The while loop(iterates i) is nested inside the for loop(iterates j). Hence for every value of j in the outer loop, the inner loop(i) iterates for t_j times.

t_j = (number of times while loop iterates for each j). Hence total cost overall would be cost summation for all j iterations which is sigma{for all j=2..N}(t_j)