Triangular faces of the order and chain polytope of
a maximal ranked poset

(a) (“If”) From Lemma 2.1(a), if $I=\emptyset$ and $J$ is connected, then ${\rm conv}(\{\rho(I),\rho(J)\})$ forms an edge of $\mathcal{O}(P)$. Since $|{\rm max}(J)|\geq 2$, ${\rm max}(I)\,\triangle\,{\rm max}(J)={\rm max}(J)$ is disconnected. Thus, by Lemma 2.1(b), ${\rm conv}(\{\rho({\rm max}(I)),\rho({\rm max}(J))\})$ does not form an edge of $\mathcal{C}(P)$.
(“Only if”) According to Lemma 2.1, $\{I,J\}\in E^{*}_{\mathcal{O}(P)}$ if and only if $I\subset J$, $J\setminus I$ is connected, and ${\rm max}(I)\,\triangle\,{\rm max}(J)$ is disconnected in $P$. If ${\rm max}(I)\triangle{\rm max}(J)$ is disconnected, then either ${\rm max}(I)=\emptyset$ and $|{\rm max}(J)|\geq 2$, or ${\rm max}(I),{\rm max}(J)$ are subsets of $P_{k}$, where $0\leq k\leq n$, since $P$ is a maximal ranked poset. If ${\rm max}(I),{\rm max}(J)\subset P_{k}$, then ${\rm max}(I)\;\triangle\;{\rm max}(J)={\rm max}(J)\setminus{\rm max}(I)=J\setminus I$, since $I\subset J$, namely ${\rm max}(I)\subset{\rm max}(J)$. Since $J\setminus I$ is connected and ${\rm max}(I)\;\triangle\,{\rm max}(J)$ is disconnected, this leads to a contradiction. Hence, we have ${\rm max}(I)=\emptyset$ and $|{\rm max}(J)|\geq 2$, as desired.

(b) (“If”) If there exists some $\ell$ such that $A=P_{\ell-1}$, $B\subset P_{\ell}$, and $|B|\geq 2$, where $1\leq\ell\leq n$, then $A\triangle B$ is connected. Then, from Lemma 2.1(b), ${\rm conv}(\{\rho(A),\rho(B)\})$ forms an edge of $\mathcal{C}(P)$. Furthermore, $\langle A\rangle\subset\langle B\rangle$ and $\langle B\rangle\setminus\langle A\rangle$ is disconnected, because $\langle B\rangle\setminus\langle A\rangle=B$ and $|B|\geq 2$. Hence, by Lemma 2.1(a), ${\rm conv}(\{\rho(\langle A\rangle),\rho(\langle B\rangle))$ does not form an edge of $\mathcal{O}(P)$.
(“Only if”) According to Lemma 2.1, $\{A,B\}\in E^{*}_{\mathcal{C}(P)}$ if and only if the following conditions hold:

• •

  $A\triangle B$ is connected in $P$ ;

• •

  $\langle A\rangle\setminus\langle B\rangle\neq\emptyset$ and $\langle B\rangle\setminus\langle A\rangle\neq\emptyset$ or $\langle A\rangle\subset\langle B\rangle$ and $\langle B\rangle\setminus\langle A\rangle$ is disconnected in $P$.

If $\langle A\rangle\setminus\langle B\rangle\neq\emptyset$ and $\langle B\rangle\setminus\langle A\rangle\neq\emptyset$, then there exists some $k$ such that $A,B\subset P_{k}$ and $A\setminus B\neq\emptyset$ and $B\setminus A\neq\emptyset$, where $0\leq k\leq n$, because $P$ is a maximal ranked poset. This contradicts $A\triangle B$ is connected. Thus, assuming $\langle A\rangle\subset\langle B\rangle$ and $\langle B\rangle\setminus\langle A\rangle$ is disconnected, there exists some $\ell$ such that $\langle B\rangle\setminus\langle A\rangle\subset P_{\ell}$ and $|\langle B\rangle\setminus\langle A\rangle|\geq 2$. In this case, either $A,B\subset P_{\ell}$ or $A=P_{\ell-1},B\subset P_{\ell}$ holds, but $A,B\subset P_{\ell}$ contradicts $A\triangle B$ is connected. Hence, there exists some $\ell$ such that $A=P_{\ell-1},B\subset P_{\ell}$ and $|\langle B\rangle\setminus\langle A\rangle|=|B|\geq 2$, where $1\leq\ell\leq n$. ∎

From Lemma 2.2, the set of triples $\{\rho(I),\rho(J),\rho(K)\}$ where $\{I,J,K\}\in\Delta_{\mathcal{O}(P)}\setminus\Delta^{*}_{\mathcal{O}(P)}$, corresponds to the set of triangles in the graph obtained by removing the all edges $\{\rho(I_{1}),\rho(I_{2})\}$, where $\{I_{1},I_{2}\}\in E^{*}_{\mathcal{O}(P)}$ from the $1$-skeleton of ${\mathcal{O}}(P)$. Similarly, the set of triples $\{\rho(A),\rho(B),\rho(C)\}$ where $\{A,B,C\}\in\Delta_{\mathcal{C}(P)}\setminus\Delta^{*}_{\mathcal{C}(P)}$, corresponds to the set of triangles in the graph obtained by removing the all edges $\{\rho(A_{1}),\rho(A_{2})\}$, where $\{A_{1},A_{2}\}\in E^{*}_{\mathcal{C}(P)}$ from the $1$-skeleton of ${\mathcal{C}}(P)$. Since these graphs are clearly isomorphic, the number of triangles in them coincides. ∎

From Lemma 2.2(a) and Lemma 3.1(a), the following conditions hold for $\{I,J,K\}\in\Delta^{*}_{\mathcal{O}(P)}$, where $I$, $J$, and $K$ be pairwise distinct poset ideals of $P$ :

• •

  $I=\emptyset$, $J\subset K$ and $J$, $K$, and $K\setminus J$ are connected in $P$;

• •

  $|{\rm max}(J)|\geq 2$ or $|{\rm max}(K)|\geq 2$.

Let $P$ has rank $n$. Since $P$ is a maximal ranked poset, we may assume that ${\rm max}(J)\subset P_{j}$ and ${\rm max}(K)\subset P_{k}$, where $0\leq j\leq k\leq n$. We define the map $\varphi\;:\;\Delta^{*}_{\mathcal{O}(P)}\rightarrow\Delta^{*}_{\mathcal{C}(P)}$ by

We will show that the map is an injection.

Case 1. $|{\rm max}(J)|\geq 2.$
Since $J$ is connected, we may assume that $j\geq 1$. Now, $P_{j-1}$, ${\rm max}(J)$ and ${\rm max}(K)$ are pairwise distinct antichains of $P$. As $P$ is a maximal ranked poset, $P_{j-1}\triangle{\rm max}(J)$ and $P_{j-1}\triangle{\rm max}(K)$ are connected. If $j<k$, then ${\rm max}(J)\triangle{\rm max}(K)$ is connected. If $j=k$, then ${\rm max}(J)\;\triangle\;{\rm max}(K)={\rm max}(K)\setminus{\rm max}(J)=K\setminus J,$ because $J\subset K$, namely ${\rm max}(J)\subset{\rm max}(K)$. Since $K\setminus J$ is connected, ${\rm max}(J)\triangle{\rm max}(K)$ is also connected. Furthermore, since $|{\rm max}(J)|\geq 2$, from Lemma 3.1(b), we have $\{P_{j-1},{\rm max}(J)\}\in E^{*}_{\mathcal{C}(P)}$, implying $\{P_{j-1},{\rm max}(J),{\rm max}(K)\}\in\Delta^{*}_{\mathcal{C}(P)}$.

In Case 1 or Case 2, the map is clearly injective. In Case 3, let $\{\emptyset,J,K\}$ and $\{\emptyset,J^{\prime},K^{\prime}\}$ belong to $\Delta^{*}_{\mathcal{O}(P)}$. We note the following holds:

• •

  ${\rm max}(J^{\prime})\subset P_{j^{\prime}}$ and ${\rm max}(K^{\prime})\subset P_{k^{\prime}}$, where $0\leq j^{\prime}\leq k^{\prime}\leq n$;

• •

  $|{\rm max}(J^{\prime})|=1$ and $k^{\prime}-j^{\prime}=1$.

Suppose that $P_{k-1}=P_{k^{\prime}-1}$, ${\rm min}(K\setminus J)={\rm min}(K^{\prime}\setminus J^{\prime})$, and ${\rm max}(K)={\rm max}(K^{\prime})$. Then $K=K^{\prime}$. Since $k-j=k^{\prime}-j^{\prime}=1$, ${\rm max}(K\setminus J)={\rm max}(K)={\rm max}(K^{\prime})={\rm max}(K^{\prime}\setminus J^{\prime})$. Thus ${\rm max}(K\setminus J)={\rm max}(K^{\prime}\setminus J^{\prime})$. Now, we have ${\rm min}(K\setminus J)={\rm min}(K\setminus J^{\prime})$ and ${\rm max}(K\setminus J)={\rm max}(K\setminus J^{\prime})$ by $K=K^{\prime}$. Since $K\setminus J={\rm min}(K\setminus J)\cup{\rm max}(K\setminus J)$, it follows that $K\setminus J=K\setminus J^{\prime}$, namely $J=J^{\prime}$, as desired. ∎

(“If”) The poset $P$ does not contain an $X$-poset from as a subposet if and only if $\mathcal{O}(P)$ and $\mathcal{C}(P)$ are unimodularly equivalent by [4, Theorem 2.1]. Thus, the $1$-skeleton of $\mathcal{O}(P)$ and that of $\mathcal{C}(P)$ are isomorphic as finite graphs. In particlar, the number of triagles in those $1$-skeletons coincides. Therefore $|\Delta_{\mathcal{O}(P)}|=|\Delta_{\mathcal{C}(P)}|$.
(“Only if”) Let $P$ has rank $n$. Suppose that the poset $P$ contains an $X$-poset from as a subposet. Let the $X$-poset be $\{a,b,c,d,e\}$, where each element is pairwise distinct such that (i) $a$ and $b$ are incomparable, (ii) $d$ and $e$ are incomparable, and (iii) $a<c<d$, $b<c<e$. We may assume that $d,e\in P_{s}$, $c\in P_{s-1}$ and $a,b\in P_{s-t}$, where $2\leq s\leq n$ and $2\leq t\leq s$. Now, $P_{s-t}$, $P_{s-1}$, and $P_{s}$ are pairwise distinct antichains. As $P$ is a maximal ranked poset, $P_{s-t}\triangle P_{s-1}$, $P_{s-1}\triangle P_{s}$, and $P_{s-t}\triangle P_{s}$ are connected in $P$. Since $|P_{s}|\geq 2$, from Lemma 3.1(b), $\{P_{s-1},P_{s}\}\in E^{*}_{\mathcal{C}(P)}$, implying $\{P_{s-t},P_{s-1},P_{s}\}\in\Delta^{*}_{\mathcal{C}(P)}$. We will show that there does not exist any $\{I,J,K\}\in\Delta^{*}_{\mathcal{O}(P)}$ such that $\varphi(\{I,J,K\})=\{P_{s-t},P_{s-1},P_{s}\}$, where $\varphi$ is the injection obtained in the proof of Lemma 3.4.

By the above arguments, the map $\varphi$ turns out to be not surjective. Consequently, it follows that $|\Delta^{*}_{\mathcal{O}(P)}|<|\Delta^{*}_{\mathcal{C}(P)}|$. From Lemma 3.3, we establish $|\Delta_{\mathcal{O}(P)}|<|\Delta_{\mathcal{C}(P)}|$, as desired. ∎