Relations on Generalized Degree Sequences

TL;DR

This paper extends the concept of degree sequences from graphs to simplicial posets and polyhedral complexes, exploring their properties, relations, and connections to known invariants like the f-vector.

Contribution

It introduces and analyzes arbitrary face-to-flag degree sequences, generalizing existing notions and establishing linear relations and inequalities.

Findings

Proves linear relations between flag degree sequences.

Recovers Stanley's f-vector inequality for simplicial posets.

Provides a framework for understanding degree sequences in higher-dimensional complexes.

Abstract

We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f-vector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an f-vector inequality on simplicial posets first shown by Stanley.