A cycle lemma for permutation inversions
The permutations of the multiset \(\{ 1^{m_1},2^{m_2}\}\) of size \(n=m_1+m_2\) can be enumerated by the number of inversions. For \(d=\gcd(m_1,m_2)\), the authors give a combinatorial proof that the remainders modulo \(n/d\) of the numbers of inversions are equally distributed, and the remainders modulo \(m_1/d\) of the numbers of inversions of permutations whose first term is 1 are equally distributed. These results provide combinatorial interpretations of some properties of the Gaussian binomial coefficients.
scientific article; zbMATH DE number 1838971