Effective poset inequalities

TL;DR

This paper investigates and generalizes inequalities related to linear extensions of posets, providing effective proofs, complexity insights, and extending results to specialized poset classes.

Contribution

It introduces new generalized inequalities, injective proofs with complexity implications, and extends existing inequalities to specific poset structures.

Findings

Generalized Bj"orner--Wachs inequality to order polynomials and q-analogues

Provided an injective proof of Sidorenko inequality with #P complexity

Extended Sidorenko inequality to posets with small chain intersections

Abstract

We explore inequalities on linear extensions of posets and make them effective in different ways. First, we study the Bj\"orner--Wachs inequality and generalize it to inequalities on order polynomials and their $q$-analogues via direct injections and FKG inequalities. Second, we give an injective proof of the Sidorenko inequality with computational complexity significance, namely that the difference is in $\#P$. Third, we generalize the Sidorenko inequality to posets with small chain intersections and give complexity theoretic applications.