Find the coefficient $x^{14}$ in the $(1+x)(1+x+x^2)(1+x+x^2+x^3)\dots(1+x+\dots+x^{5})$.

Question

This is corresponding to finding no. of permutation of $\{1, 2, 3, 4, 5, 6\}$ having exactly $14$ inversions.

Answer 1

First of all, the product you have written down is incorrect. It should be $$ (1+x)(1+x+x^2)\cdots (1+x+\cdots +x^5), $$ at least if you want to count the number of inversions.

The coefficients of this polynomial are symmetric (i.e. the coefficient of $x^n$ is the same as the coefficient of $x^{15-n}$). Therefore the coefficient of $x^{14}$ is the same as the coefficient of $x$, which is $5$.

There is an easy combinatorial explanation: we have to leave out one inversion, and there are 5 ways to pick the inversion you leave out: it is one of $$ (12), (23), (34), (45), (56). $$