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Order and chain polytopes of maximal ranked posets

Ibrahim Ahmad, Ghislain Fourier, Michael Joswig

TL;DR

The paper addresses the problem of comparing the $f$-vectors of order and chain polytopes by focusing on chain-order polytopes that interpolate between them for maximal ranked posets. It introduces a detailed face normal form and constructs an injective map $\psi$ between faces across consecutive decompositions, enabling a pointwise comparison of face counts and establishing $f$-vector monotonicity. The main contribution is an independent proof of the Hibi–Li conjecture for maximal ranked posets, achieved by a combinatorial analysis of faces and a constructive injection between chain-order polytopes, complemented by a computational framework to verify and illustrate the results on concrete examples. The work deepens understanding of the facet–face structure of chain-order polytopes and provides a practical route to verify $f$-vector monotonicity in a broad class of posets, with potential extensions to wider poset families.

Abstract

The order and chain polytopes, introduced by Richard P. Stanley, form a pair of Ehrhart equivalent polytopes associated to a given finite poset. A conjecture by Takayuki Hibi and Nan Li states that the $f$-vector of the chain polytope dominates the $f$-vector of the order polytope. In this paper we prove a stronger form of that conjecture for a special class of posets. More precisely, we show that the $f$-vectors increase monotonically over an admissible family of chain-order polytopes for such posets.

Order and chain polytopes of maximal ranked posets

TL;DR

The paper addresses the problem of comparing the -vectors of order and chain polytopes by focusing on chain-order polytopes that interpolate between them for maximal ranked posets. It introduces a detailed face normal form and constructs an injective map between faces across consecutive decompositions, enabling a pointwise comparison of face counts and establishing -vector monotonicity. The main contribution is an independent proof of the Hibi–Li conjecture for maximal ranked posets, achieved by a combinatorial analysis of faces and a constructive injection between chain-order polytopes, complemented by a computational framework to verify and illustrate the results on concrete examples. The work deepens understanding of the facet–face structure of chain-order polytopes and provides a practical route to verify -vector monotonicity in a broad class of posets, with potential extensions to wider poset families.

Abstract

The order and chain polytopes, introduced by Richard P. Stanley, form a pair of Ehrhart equivalent polytopes associated to a given finite poset. A conjecture by Takayuki Hibi and Nan Li states that the -vector of the chain polytope dominates the -vector of the order polytope. In this paper we prove a stronger form of that conjecture for a special class of posets. More precisely, we show that the -vectors increase monotonically over an admissible family of chain-order polytopes for such posets.
Paper Structure (27 sections, 13 theorems, 51 equations, 5 figures, 2 algorithms)

This paper contains 27 sections, 13 theorems, 51 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Face normal form for chain-order polytope
  4. Proof of the main theorem
  5. Construction: Preliminaries
  6. Face partition:
  7. Zero elements:
  8. Chain elements:
  9. Construction: Generic Case
  10. All elements of $Y^{{k+1}}$ are in singletons in $\pi$
  11. Not all elements of $Y^{{k+1}}$ are in singletons blocks in $\pi$
  12. Construction: Degenerate Case
  13. All elements of $Y^{{k+1}}$ are in singletons in $\pi$
  14. Not all elements of $Y^{{k+1}}$ are in singletons in $\pi$
  15. Assume that the height of $K$ is at least two
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $P$ be a poset. Then every non-empty face $F\subseteq\mathcal{O}(P)$ can be identified with a partition $\pi_F$ of $\hat{P}$ that is connected, $\hat{P}$-compatible such that $\hat{0}$ and $\hat{1}$ are in different blocks. Further, the map is an affine isomorphism. In particular, we have $\dim(F)=|\pi_F|-2$.

Figures (5)

  • Figure 1: Normal form of a face of $\mathcal{O}_{C,O}((5,2,1,4,2,3))$ with $k=3$ of codimension $5$. The elements in red belong to $F_{\mathrm{eq},i}$ for respective $i$, the element in blue belongs to $F_{0,2}$ and the green block is a block of cardinality $3$ in a face partition of $O\cup\{\hat{1}\}$. Elements in $O$ that are not marked are in singletons.
  • Figure 2: Example of construction in Case \ref{['Case-Gen-Non-Single']} when $F$ already uses a chain and $K$ is of height $1$. The orange blocks are blocks of the face partitions of $O\cup\{\hat{1}\}$ and $O'\cup\{\hat{1}\}$, respectively. For the remaining color-coding, see figure \ref{['ex:running']}.
  • Figure 3: Example of construction in Case \ref{['Case-Deg-Height-1-Max-Diff']}. For color-coding, see figure \ref{['ex:running']}.
  • Figure 4: Face numbers of the order and chain polytopes of $P_\tau$ for $n=\sum \tau_i = 10$, omitting trivial cases
  • Figure 5: $f$-vector of the running example $\tau=(5,2,1,4,2,3)$

Theorems & Definitions (24)

  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3: FFLP
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 14 more