Expectations of Tutte-related functions of random ranked sets with multiplicities

TL;DR

This paper introduces models that connect expectations of counting functions in ranked sets with multiplicities to multivariate Tutte polynomials, extending classical results and providing new probabilistic interpretations.

Contribution

It develops two models that generalize existing formulas to compute expectations of various invariants in ranked sets with multiplicities, including new interpretations of Tutte polynomials.

Findings

Expectations of counting functions are given by multivariate Tutte polynomials.

Generalization of Kung's convolution formula to ranked sets with multiplicities.

New convolution-like formula for Ehrhart polynomials of lattice zonotopes.

Abstract

Employing two models, we show that various counting functions of a random variable defined by restriction or contraction of a ranked set with multiplicity (e.g., classical and arithmetic matroids) have expectations given by the corresponding multivariate Tutte polynomial. The first model is based on a generalization of a convolution formula of Kung (2010), extending from matroids to ranked sets with multiplicities. This model enables us to compute the expectations of many familiar polynomials, such as the chromatic, flow and Ehrhart polynomials, generalizing the classical results of Welsh (1996) on random graphs. The second model is designed to compute the expectations of invariants that are generally not evaluations of the polynomials mentioned above, such as the number of connected components of an intersection of hypersurfaces in an abelian Lie group arrangement, and the number of lattice points in a half-open zonotope. In particular, both models yield new probabilistic interpretations of the arithmetic Tutte polynomial and $G$-Tutte polynomial. A simple, but seems to be new convolution-like formula for the Ehrhart polynomials of lattice zonotopes will also be given.