Linear inequalities for flags in graded posets

TL;DR

This paper characterizes the convex cone of flag f-vectors in graded posets, providing facet inequalities, exploring extreme rays, and describing the strongest inequalities for low-rank cases.

Contribution

It establishes the polyhedral structure of the cone of flag f-vectors, identifies facet inequalities via antichains of intervals, and analyzes extreme rays under convolution.

Findings

The cone of flag f-vectors is polyhedral.

Facet inequalities correspond to antichains of intervals, counted by Catalan numbers.

Convolution often assigns extreme rays to pairs of extreme rays.

Abstract

The closure of the convex cone generated by all flag $f$-vectors of graded posets is shown to be polyhedral. In particular, we give the facet inequalities to the polar cone of all nonnegative chain-enumeration functionals on this class of posets. These are in one-to-one correspondence with antichains of intervals on the set of ranks and thus are counted by Catalan numbers. Furthermore, we prove that the convolution operation introduced by Kalai assigns extreme rays to pairs of extreme rays in most cases. We describe the strongest possible inequalities for graded posets of rank at most 5.