dinv

DinV, short for diagonal inversion, is a statistic used in algebraic combinatorics to measure certain inversions in combinatorial objects such as Dyck paths and parking functions. It counts pairs of elements that lie on the same or adjacent diagonals in a diagram associated with the object, in a way that generalizes the ordinary inversion count for permutations.

In Dyck-path models, dinv can be defined using the area sequence or by tracking pairs of steps that form a diagonal inversion; in parking-function models it is defined on labelled structures and records how labels are arranged with respect to the diagonal. Different but equivalent definitions exist across these related models, reflecting the same underlying combinatorial idea.

DinV is commonly studied together with the area statistic. The two statistics form a bi-graded measure whose generating function produces the q,t-Catalan polynomials and, in many formulations, realizes the character of the diagonal harmonic module. This connection is a central part of the shuffle theorem and related work by Haglund, Haiman, Loehr, Remmel, and others.

Computing dinv typically involves applying the chosen definition to the object and counting qualifying pairs of steps, cells, or labels. The statistic has several variants depending on the combinatorial model and is a standard tool in the study of diagonal harmonics and Macdonald polynomial theory.

The term dinv stands for diagonal inversion and has become a standard concept in this area of combinatorics. See also area statistic, q,t-Catalan numbers, diagonal harmonics, and Macdonald polynomials.