Inversion of a word

For a word $w=w_1\cdots w_n$ over a totally ordered alphabet, an inversion of a word is a pair of positions $i<j$ with $w_i>w_j$. Equal symbols contribute no inversion of a word. In baba, with $a<b$, the pairs are $(1,2),(1,4),(3,4)$, so the word inversion number is three. Counting these pairs records distance from sorted order even when letters repeat; replacing the letters by distinct ordered labels gives the inversion of a permutation case.

The number of inversions of a word: pairs of positions $i<j$ with $w_i>w_j$. Unlike an inversion of a permutation, this definition permits repeated letters in the word.