Problem: How many permutations of $\{1,2,3,4,5,6,7,8\}$ have exactly $15$ inversions?
Definitions:
- Inversion sequence: Let $i_1i_2i_3..i_n$ be a permutation of the set $\{1,2,3,...,n\}$. The sequence of numbers $a_1a_2a_3...a_n$ is called the inversion sequence of the permutation $i_1i_2i_3..i_n$ if $a_j$ for $1\leq j\leq n$ equals the number of integers that precede $j$ in the permutation but are greater than $j$. For example for the set $\{1,2,3,4\}$ and the permutation $3214$, the inversion sequence is $2,1,0,0.$
- Number of inversions: If the permutation $i_1i_2i_3..i_n$ has the inversion sequence $a_1a_2a_3...a_n$ then the number of inversions $k=a_1+a_2+a_3+...+a_n.$
My Attempt: Since each inversion sequence yields a unique permutation, counting the number of permutations is identical to counting the number of inversion sequences $a_1a_2a_3...a_8$ such that $a_1+a_2+a_3+...+a_8=15.$ Now there is also a constraint on each variable by Definition 1 and therefore $$0\leq a_i\leq 8-i.$$
I am having trouble in figuring out an analytical expression for the number of solutions to this equation and would much appreciate if some insights are offered in this regard.