Question 2
Let $m_1,\ldots,m_n$ be positive integers, possibly repeated. For an integer sequence $A=(a_1,\ldots,a_n)$ and a permutation $w$ of this multiset, an $A$-inversion is a pair $i\lt j$ satisfying one of
$$a_i\ge w_i\gt w_j,\qquad w_j\gt a_i\ge w_i,\qquad w_i\gt w_j\gt a_i.$$
Prove that for any integer sequences $A,B$ and every positive integer $k$, the numbers of permutations having exactly $k$ $A$-inversions and exactly $k$ $B$-inversions are equal.
Proposed by Maria Monks. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.
$$a_i\ge w_i\gt w_j,\qquad w_j\gt a_i\ge w_i,\qquad w_i\gt w_j\gt a_i.$$
Prove that for any integer sequences $A,B$ and every positive integer $k$, the numbers of permutations having exactly $k$ $A$-inversions and exactly $k$ $B$-inversions are equal.
Proposed by Maria Monks. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.