\PassOptionsToPackage{hypertexnames=false}{hyperref} \documentclass[ALCO, Unicode]{cedram} \usepackage[matrix,arrow,tips,curve]{xy} \usepackage{tikz,tkz-graph} \usepackage{subcaption} \usetikzlibrary{calc,shapes,fit,positioning,arrows,automata,decorations.pathreplacing} \newcommand{\la}{\longrightarrow} \newcommand{\NN}{\mathbb{N}} \newcommand{\E}{\mathcal{E}} \newcommand{\sym}{\mathfrak{S}} \newcommand{\antiexcl}{\rotatebox[origin=c]{180}{!}} \newcommand{\OP}{\mathcal{P}} \newcommand{\Perm}{\mathcal{P}erm} \newcommand{\Com}{\mathcal{C}om} \newcommand{\Prelie}{\mathcal{P}re\mathcal{L}ie} \newcommand{\Lie}{\mathcal{L}ie} \newcommand{\cover}{\lessdot} \newcommand{\Flyn}{\mathcal{FL}yn} \newcommand{\clr}{\mathbf{color}} \newcommand{\ISF}{\mathcal{ISF}} \newcommand{\SF}{\mathcal{SF}} \newcommand{\bpi}{\pmb{\pi}} \newcommand{\balpha}{\pmb{\alpha}} \newcommand{\tre}[1]{{\color{red}{#1}}} \newcommand{\tbl}[1]{{\color{blue}{#1}}} \newcommand{\sort}{\mathrm{sort}} \newcommand{\sgn}{\mathrm{sgn}} \begin{DefTralics} \newcommand{\Perm}{\mathcal{P}erm} \newcommand{\Com}{\mathcal{C}om} \newcommand{\Prelie}{\mathcal{P}re\mathcal{L}ie} \end{DefTralics} \title[Whitney twins, Whitney duals, and operadic partition posets]{Whitney twins, Whitney duals, and operadic partition posets} \author{\firstname{Rafael}\nobreakauthor\middlename{S.}\nobreakauthor\lastname{Gonz\'alez D'Le\'on}} \address{Loyola University Chicago\\ Department of Mathematics and Statistics\\ Chicago, IL 60660 (USA)} \email{rgonzalezdleon@luc.edu} \urladdr{http://dleon.combinatoria.co} \author{\firstname{Joshua}\nobreakauthor\lastname{Hallam}} \address{Loyola Marymount University\\ Department of Mathematics, Statistics and Data Science\\ Los Angeles, CA 90045 (USA)} \email{Joshua.Hallam@lmu.edu} \urladdr{https://jhallam.lmu.build/} \author{\firstname{Yeison}\nobreakauthor\middlename{A.}\nobreakauthor\lastname{Quiceno D.}} \address{University of Florida\\ Department of Statistics\\ Gainesville, FL 32611 (USA)} \email{yeison.quicenodu@ufl.edu} \thanks{The first author was partially supported by an AMS-Simons Research Enhancement Grant for Primarily Undergraduate Institution Faculty.} \keywords{Whitney numbers, Whitney twins, Whitney duality, operadic partition posets, lexicographic shellability, PBW bases, Koszul duality} \subjclass{01A35, 14A15, 11L05} \begin{document} \begin{abstract} In this article we address the question of uniqueness posed by the results on edge labelings and Whitney duality, recently developed by the first two authors. We do this by giving examples of families of posets with multiple Whitney duals. More precisely, we study edge labelings for the families of posets of pointed partitions $\Pi_n^{\bullet}$ and weighted partitions $\Pi_n^{w}$ which are associated to the operads $\Perm$ and $\Com^2$ respectively. The first author and Wachs proved that these two families of posets share the same pair of sequences of Whitney numbers. We find EW-labelings for $\Pi_n^{\bullet}$ and $\Pi_n^{w}$ and use them to show that they also share multiple non-isomorphic Whitney dual posets. Along the way, we find two new EL-labelings for $\Pi_n^\bullet$ answering a question of Chapoton and Vallette about the existence of such a labeling. Using these EL-labelings of $\Pi_n^\bullet$, and an EL-labeling of $\Pi_n^w$ introduced by the first author and Wachs, we give combinatorial descriptions of bases for the operads $\Prelie, \Perm,$ and $\Com^2$. We also show that the bases for $\Perm$ and $\Com^2$ are PBW bases. \end{abstract} \maketitle \setcounter{secnumdepth}{3} \section{Introduction} To a finite graded \emph{poset} (partially ordered set) $P$ with a minimal element (denoted~$\hat{0}$ throughout) we can associate a pair of sequences of integers $\{w_k(P)\}_{k\ge 0}$ and $\{W_k(P)\}_{k\ge 0}$ known as the \emph{Whitney numbers of the first and second kind} respectively. These two sequences are poset invariants and encode relevant information in areas where partially ordered structures arise naturally. For example, Whitney showed in~\cite{Whitney1932} that the coefficients of the chromatic polynomial of a graph are the Whitney numbers of the first kind of a poset one can associate to a graph (its bond lattice). The Whitney numbers of the first kind keep track of the M\"obius function at each rank level and the Whitney numbers of the second kind keep track of the number of elements at each rank level. \subsection{Whitney-realizable and dualizable sequences} In~\cite{GonzalezHallam2021}, the first and second authors introduced the concept of a Whitney dual of a graded poset $P$ with a $\hat{0}$. We say that two graded posets $P$ and $Q$ are \emph{Whitney duals} if, after taking absolute values, the sequences of Whitney numbers of the first and second kind of $P$ are equal to the sequences of Whitney numbers of the second and first kind of $Q$. That is, the Whitney numbers of $P$ and $Q$ are swapped with respect to one another. In ~\cite{GonzalezHallam2021}, the authors also defined a new type of poset edge labeling, which is called an EW-labeling (or Whitney labeling). The authors show that these labelings provide a sufficient condition for the existence of a Whitney dual for any graded poset $P$ admitting such a labeling. Moreover, they describe an explicit construction of the Whitney dual associated to a given EW-labeling. One can readily observe from the definition, that nothing prevents the existence of multiple Whitney duals to a graded poset $P$. Hence, the concept of Whitney duality is more precisely a duality between the sequences of numbers involved rather than a duality between posets. Recently, there has been a great interest in the Whitney numbers of geometric lattices in relation to the properties of unimodality and the stronger property of log-concavity of positive integer sequences. In particular, it was shown in~\cite{AdiprasitoHuhKatz2018} that the absolute values of the Whitney numbers of the first kind of geometric lattices form a log-concave sequence, which was a well-known conjecture by Heron, Rota, and Welsh. It has been conjectured that this is also the case for the Whitney numbers of the second kind (see~\cite[page 289]{Welsh1976}). Another geometric property of the latter set of numbers was recently established in ~\cite{BradenHuhMatherneProudfootWang2022}. These results answer long-standing questions about the structure of Whitney numbers for geometric lattices and open a pathway for a technique to prove these properties for an arbitrary integer sequence, giving relevance to the question of realizability. We say that a pair of nonnegative integer sequences $(\{a_k\}_{k\ge 0},\{b_k\}_{k\ge 0})$ is \emph{Whitney-realizable} if there exists a poset $P$ such that $(\{|w_k(P)|\}_{k\ge 0},\{W_k(P)\}_{k\ge 0})=(\{a_k\}_{k\ge 0},\{b_k\}_{k\ge 0})$. We will call two posets $P$ and $Q$ \emph{Whitney twins} if they realize the same pair of sequences. We say that a Whitney-realizable pair is \emph{Whitney-dualizable} if~$(\{b_k\}_{k\ge 0},\{a_k\}_{k\ge 0})$ is also Whitney-realizable. Determining which pairs of nonnegative integer sequences $(\{a_k\}_{k\ge 0},\{b_k\}_{k\ge 0})$ are Whitney-realizable or Whitney-dualizable both seem to be challenging questions. In this article, we present results related to the non-uniqueness of Whitney realizations and dualizations of a pair $(\{a_k\}_{k\ge 0},\{b_k\}_{k\ge 0})$ by finding and exploring the algebraic and combinatorial consequences of EW-labelings on two families of posets which come from the theory of symmetric operads. These two particular families of posets are associated to the permutative operad $\Perm$ and to the double commutative operad $\Com^2$. \subsection{Operadic posets and EL/CL-labelings} In~\cite{Vallette2007}, Vallete defined a family of partition posets $\Pi_n^{\OP}$ associated to a basic-set quadratic operad $\OP$. These posets are an operadic generalization of the poset of set partitions $\Pi_n$ ordered by refinement. There, the author shows that the top cohomology $\sym_n$-modules $H^{top}(\Pi_n^{\OP})$ are, up to tensoring with the sign representation, equal to the Koszul dual cooperad $\OP^{\antiexcl}$ to~$\OP$. He also shows that the Cohen-Macaulay property of the maximal intervals of $\Pi_n^{\OP}$ is equivalent to the Koszul property of $\OP$ and~$\OP^{\antiexcl}$. Hence, the application of combinatorial techniques on the family $\Pi_n^{\OP}$ is relevant in determining the algebraic properties of~$\OP$ and $\OP^{\antiexcl}$. One such technique is the theory of lexicographic shellability for posets introduced by Bj\"orner~\cite{Bjorner1980} and further developed by Bj\"orner and Wachs in~\cite{BjornerWachs1982, BjornerWachs1983} (see also~\cite{BjornerWachs1996,BjornerWachs1997}). The main idea behind the theory of lexicographic shellability is that the maximal intervals of a poset $P$ which admit a type of edge labeling, known as an EL-labeling (or a CL-labeling in more generality), are Cohen-Macaulay. Finding an EL or CL-labeling for a poset $\Pi_n^{\OP}$ then implies under Vallette's relation that $\OP$ and $\OP^{\antiexcl}$ are Koszul operads. As an application of EL and CL-labelings for partition posets, Bellier-Mill{\`e}s, Delcroix-Oger, and Hoffbeck~\cite{BelliermillesDelcroixogerHoffbeck2021} showed that if an EL or CL-labeling of $\Pi_n^{\OP}$ satisfies a certain condition that they call being \emph{isomorphism-compatible}, then the operad $\OP$ has a Poincar\'{e}--Birkhoff--Witt (PBW) basis determined by the labeling. PBW bases are useful because they imply that the operads are Koszul as was shown by Hoffbeck~\cite{Hoffbeck2010} for totally ordered PBW bases and in more generality for partially ordered PBW bases in~\cite{BelliermillesDelcroixogerHoffbeck2021}. We note that the posets $\Pi_n^{\OP}$ have appeared before in a different but related context. They are relevant in finding compositional (or substitutional) inverses to species within Joyal's theory of combinatorial species (see~\cite{Joyal1981,BergeronLabelleLeroux1998}) as was shown by M\'endez and Yang in~\cite{MendezYang1991}. \subsection{Pointed and weighted partition posets} Vallette~\cite{Vallette2007} showed that the pointed partition poset $\Pi_{n}^{\bullet}$ is isomorphic to the operadic poset $\Pi_n^{\Perm}$ associated to the operad $\mathcal{P}erm$. In Section~\ref{sec:EWLabOfPointPart}, we give an EW-labeling of $\Pi_n^\bullet$ and give an explicit description of its Whitney dual in terms of pointed Lyndon forests in Section~\ref{subsection:whitney_dual_pointed}. In~\cite{ChapotonVallete2006}, Chapoton and Vallette show that the maximal intervals of $\Pi_{n}^{\bullet}$ are totally semimodular. By the results in~\cite{BjornerWachs1983}, this implies that they are also CL-shellable and hence Cohen-Macaulay. By the result in~\cite{Vallette2007} this in turn implies that $\mathcal{P}erm$, and its Koszul-dual operad $\mathcal{P}re\mathcal{L}ie$, are Koszul. The authors in~\cite{ChapotonVallete2006} leave open the question of whether or not $\Pi_{n}^{\bullet}$ admits the more restrictive property of being EL-shellable. EL-shellability and CL-shellability have been shown recently by Li~\cite{Tiansi2020} to not be equivalent in general for posets. The authors in~\cite{BelliermillesDelcroixogerHoffbeck2021} propose a possible EL-labeling of~$\Pi_{n}^{\bullet}$ and claim that this labeling has the additional property of being isomorphism-compatible. We show in Section~\ref{section:El-labeling_pointed_partition} that the proposed labeling does not satisfy the requirements for being an EL-labeling. We then provide a new EL-labeling which answers the open question in~\cite{ChapotonVallete2006}. This labeling has the same set of labels as our EW-labeling for $\Pi_{n}^{\bullet}$, but differ in how these labels are partially ordered. We show this EL-labeling is isomorphism-compatible which in turn gives a PBW basis for the~$\Perm$ operad using the results in~\cite{BelliermillesDelcroixogerHoffbeck2021}. Although our EW-labeling for $\Pi_{n}^{\bullet}$ is not directly an EL-labeling, we show that reversing the order on the labels gives an EL-labeling for its \emph{order dual} $(\Pi_{n}^{\bullet})^{*}$, which is the poset obtained from $\Pi_{n}^{\bullet}$ by reversing all order relations (unrelated with the Whitney dual despite the fact that both use the word ``dual''). This provides a second answer to the open question in~\cite{ChapotonVallete2006}. We also show that the former EL-labeling for $\Pi_{n}^{\bullet}$ is isomorphism-compatible, giving us a PBW bases for~$\Perm$. In~\cite{DotsenkoKhoroshkin2007}, Dotsenko and Khoroshkin introduced the weighted partition poset $\Pi_n^w$. They showed that $\Pi_n^w$ is isomorphic to the poset $\Pi_n^{\Com^2}$ associated to the operad $\Com^2$ of algebras with two totally commutative products. The combinatorial and homological properties of $\Pi_n^w$ were extensively studied by Gonz\'alez D'Le\'on and Wachs in~\cite{GonzalezDleonWachs2016}. In their study, the authors introduced an EL-labeling for $\Pi_n^w$. In Section~\ref{sec:EWforWeighted} we prove that this labeling is an EW-labeling and hence $\Pi_n^w$ has a Whitney dual. In Section~\ref{section:WhitneyDualOfWeighted}, we give an explicit description of this Whitney dual in terms of bicolored Lyndon forests. We also show in Section~\ref{sec:algConseq} that this labeling is isomorphism-compatible which gives a PBW basis for $\Com^2$. \subsection{Non-uniqueness of Whitney realizations} In~\cite[Section 2.4]{GonzalezDleonWachs2016} the authors show that $\Pi_n^w$ and $\Pi_n^{\bullet}$ are Whitney twins (though they do not use this terminology). Indeed as a consequence of their Theorem 2.8, Proposition 2.1, and the follow up discussion in Section 2.4 in~\cite{GonzalezDleonWachs2016}, the Whitney numbers of the first and second kind are given for all $k\ge 0$ by the sequences \begin{align*} w_k(\Pi_n^{\bullet})=w_k(\Pi_n^{w})&=(-1)^{k}\binom{n-1}{k}n^{k}\\ W_k(\Pi_n^{\bullet})=W_k(\Pi_n^{w})&=\binom{n}{k}(n-k)^{k}. \end{align*} This already implies the non-uniqueness of realizations for a Whitney-realizable sequence. We show that the Whitney duals constructed with the EW-labelings for~$\Pi_n^\bullet$ and~$\Pi_n^w$ are not isomorphic for $n\geq 4$. Since they have the same Whitney numbers of both kinds, we get multiple non-isomorphic Whitney duals for both $\Pi_n^\bullet$ and $\Pi_n^w$, implying further the non-uniqueness of dual realizations of Whitney-dualizable sequences. We also show that there is a third family $\SF_n$ of Whitney duals to $\Pi_n^\bullet$ and $\Pi_n^w$ which for $n\ge 3$ is not isomorphic to any of the Whitney duals discussed before. The family $\SF_n$ is also shown in future work by the first two authors to be associated with a more general type of Whitney labeling. The three non-isomorphic families of Whitney dual posets to $\Pi_n^\bullet$ and $\Pi_n^w$ also constitute a new example of the non-uniqueness of Whitney realizations. \subsection{Organization of this article} The rest of the article is structured as follows. In Section~\ref{sec:EWLab} we review EW-labelings and EL-labelings, and we describe the labelings of $\Pi_n^w$ and $\Pi_n^\bullet$. In Section~\ref{sec:CombDescriptOfWhitneyDual}, we give explicit descriptions of the Whitney duals of~$\Pi_n^w$ and~$\Pi_n^\bullet$. In Section~\ref{sec:algConseq} we consider the algebraic consequences of these labelings. Specifically we use these labelings to describe bases for $\Prelie$, $\Perm$, and $\Com^2$, the latter two in particular being PBW bases. In Section~\ref{sec:twins}, we discuss the non-uniqueness of Whitney realizations using our results for $\Pi_n^w$ and $\Pi_n^\bullet$ and their associated Whitney duals. Some results in this work have been announced as part of the third author's master's thesis in~\cite{QuicenoDuran2020}. \section{EW-labelings}\label{sec:EWLab} In this section we describe three edge labelings: one for the weighted partition poset, which was introduced already in~\cite{GonzalezDleonWachs2016}, and two new edge labelings for the pointed partition poset. The edge labeling for the weighted partition poset, was shown in~\cite{GonzalezDleonWachs2016} to be an EL-labeling and here we show that it is also an EW-labeling. Of the two labelings for the pointed partition poset, one is an EW, which we also show is a dual EL-labeling, and the second is an EL-labeling (but not an EW-labeling). We show that the two labelings have the same sets of words of labels, however the labels come from two different partial orders. Our main use of these labelings is three-fold: constructing Whitney duals for the two posets, understanding their homotopy type and cohomology of the respective order complexes, and finding PBW bases of the corresponding operads and bases for their dual (co)operads. We start with a brief review of Whitney numbers, Whitney duals, and edge labelings. \subsection{Whitney numbers and Whitney duals} We will assume some familiarity with posets. For a more in-depth review of posets as well as any undefined terms, see~\cite[Chapter 3]{Stanley2012}. For a review of poset topology see~\cite{Wachs2007}. All the posets we consider in this article will be finite, graded, and contain a minimum element which we denote by $\hat{0}$. We will use $\rho(x)$ for the rank of an element $x$. The (one-variable) \emph{M\"obius function} of a poset $P$, denoted by $\mu$, is defined recursively by $$ \mu(\hat{0}) =1 $$ and for $x\neq \hat{0}$, $$ \mu(x) = -\sum_{y2$ since the first label along the increasing chain is the smallest possible label. First, suppose that $i-1i$ or $(1,b)^u$ with $b< i$ and $u\in \{0,1\}$. By this point along $d$, $p$ is the pointed element in its block and this block contains $1$. So, all the labels at this point must have an exponent of $1$ and thus it cannot be of the form $(1,b)^0$. It also cannot be of the form $(1,b)^1$ with $b2$. If $|B_{s}|=2$ there is only one label of the form $(\min B_{s},a)^u$ where $a$ is the unique element in $B_{s}\setminus \{\min B_{s}\}$ and $u\in\{0,1\}$ is uniquely determined to be $u=0$ if $p_s=a$ or $u=1$ otherwise. In either case, all other labels in $[\hat{0},\bpi]$ are of the form $(\min B_{i},b)^u$, which are strictly smaller than $(\min B_{s},a)^u$. Now suppose that $|B_{s}|>2$ and let $b=\max B_s$. If $p_s\neq b$ then the label in the uppermost cover relation of $c_{\bpi}$ is $(\min B_{s},b)^1$ (see the proof of Proposition~\ref{proposition:ER_pointed}) which is the largest label among all labels that appear in the interval $[\hat{0},\bpi]$. Since $b$ is the largest value in its block, there is only one $\balpha'$ with this property. That is $\balpha'=\{B_1^{p_1},\dots,B_s\setminus\{b\}^{p_s},\{b\}^{b},B_{s+1}^{p_{s+1}},\dots,B_l^{p_l}\}$ which is the second to last element of $c_{\bpi}$. If $p_s= b$ then the label in the uppermost cover relation of $c_{\bpi}$ is $(\min B_{s},c)^1$ where $c=\max (B_s \setminus \{b\})$. Note that the only label of the form $(\min B_{s},a)^u$ larger than $(\min B_{s},c)^1$ is $(\min B_{s},b)^1$. But this label cannot actually appear among the cover relations $\balpha'\lessdot \bpi$ since this would indicate that $p_s\neq b$ which is not the case. Thus $(\min B_{s},c)^1$ is the largest possible label that can occur in $[\hat{0}, \bpi]$ and this also determines uniquely $\balpha'=\{B_1^{p_1},\dots,B_s\setminus\{c\}^{b},\{c\}^{c},B_{s+1}^{p_{s+1}},\dots,B_l^{p_l}\}$ which is the second to last element of $c_{\bpi}$. We conclude that $\lambda^*_\bullet$ is an EL-labeling. \end{proof} We finish this section with a remark regarding a proposed edge labeling of $\Pi_n^\bullet$ given by Bellier-Mill{\`e}s, Delcroix-Oger, and Hoffbeck in~\cite {BelliermillesDelcroixogerHoffbeck2021}.\footnote{In the ArXiv v2 version of this paper, the authors have not included this labeling anymore. We include here an explanation of why the proposed labeling is not an EL-labeling.} \begin{rema}\label{remark:proposed_EL_labeling} The authors in~\cite{BelliermillesDelcroixogerHoffbeck2021} define a labeling $\tilde \lambda:\mathcal{E}(\Pi_{n}^{\bullet})\rightarrow \NN\times \NN$ where $\NN\times \NN$ has the lexicographic order. To describe their labeling, let $\bpi \lessdot \bpi'$ be such that the two pointed blocks in $\bpi$ which were $u$-merged to get $\bpi'$ are $A_i^{q_i}$ and $A_j^{q_j}$, with $a=\min(A_i)< b=\min(A_j)$. Then define $$\tilde \lambda(\bpi \lessdot \bpi')= \begin{cases} (b,a+n-|\bpi|) & \quad \text{if }u=0,\\ (b,b+n-|\bpi|) & \quad \text{if } u=1. \end{cases}$$ With this labeling in the interval $[\hat{0},12\tilde{3}]$ of $\Pi_3^{\bullet}$, we see that the chains $\hat{0}\lessdot \tilde{1}2/\tilde {3}\lessdot 12\tilde{3}$ and $\hat{0}\lessdot 1\tilde{2}/ \tilde {3}\lessdot 12\tilde{3}$ have words of labels $(2,2)(3,2)$ and $(2,1)(3,2)$ respectively, which are both increasing in the lexicographic order on $\NN\times \NN$. This shows that $\tilde \lambda$ already fails to satisfy the requirement of the uniqueness of the increasing chain in the interval $[\hat{0},[3]^3]$. Note that this issue does not arise in the interval $[\hat{0},[3]^1]$. So $\tilde \lambda$ is an EL-labeling of at least one maximal interval of $\Pi^{\bullet}_{3}$. Moreover since $[\hat{0},[3]^1] \cong [\hat{0},[3]^3]$, their labeling shows all maximal intervals of $\Pi_{3}^\bullet$ have an EL-labeling. However, by extending this idea one can show that if $n\geq 6$, $\tilde \lambda$ is not an EL-labeling for any maximal interval of $\Pi_n^{\bullet}$. To see why, note that if $n\geq 6$, $[\hat{0}, [n]^i]$ always contains an interval of the form $[\hat{0}, [6]^j/\tilde{7}/\cdots/ \tilde{n}]$ where $j\in [6]$. If $j=4,5,6$ then the interval $[\hat{0}, 12\tilde{3}/\tilde{4}/\cdots/\tilde{n}]$ is in $[\hat{0}, [6]^j/\tilde{7}/\cdots/ \tilde{n}]$ and so we have the same problem as before. If $j=1,2,3$, then we claim the interval $[123^{j}/\tilde{4}/\dots /\tilde{n}, 123^j/45\tilde{6}/\tilde{7}\cdots/\tilde{n}]$ has two increasing chains. The chain $$ 123^{j}/\tilde{4}/\tilde{5}/\tilde{6}/\cdots /\tilde{n} \cover 123^{j}/\tilde{4}5/\tilde{6}/\cdots /\tilde{n} \cover 123^{j}/45\tilde{6}/\cdots /\tilde{n} $$ has label sequence $(5,7)(6,7)$ and the chain $$ 123^{j}/\tilde{4}/\tilde{5}/\tilde{6}/\cdots /\tilde{n} \cover 123^{j}/4\tilde{5}/\tilde{6}/\cdots /\tilde{n} \cover 123^{j}/45\tilde{6}/\cdots /\tilde{n} $$ has label sequence $(5,6)(6,7)$, both of which are increasing. It follows that $\tilde \lambda$ is not an EL-labeling in general. \end{rema} \section{Combinatorial Descriptions of Whitney duals} \label{sec:CombDescriptOfWhitneyDual} In this section we give combinatorial descriptions of the Whitney duals of $\Pi_n^w$ and $\Pi_n^\bullet$ that come from the EW-labelings we discussed in Section~\ref{sec:EWLab}. \subsection{Constructing Whitney duals}\label{sec:constructingWD} We start with a quick review of how to construct Whitney duals from EW-labelings. The full details can be found in~\cite{GonzalezHallam2021}. Note that in~\cite{GonzalezHallam2021}, the authors introduce two (isomorphic) Whitney duals that can be constructed using an EW-labeling. The first construction, which is denoted $Q_{\lambda}(P)$ in~\cite{GonzalezHallam2021}, is obtained by taking a quotient of the poset of saturated chains containing $\hat{0}$. This quotient is based on a quadratic relation on these chains. The second construction, denoted $R_{\lambda}(P)$, is a poset on ascent-free saturated chains containing $\hat{0}$ and whose order relation is given by sorting a label into an existing ascent-free chain. Here we will only use the sorting description. Let $\Lambda$ be a poset of labels and consider the following sorting algorithm on words over $\Lambda$. Given the word $w=w_1w_2\cdots w_iw_{i+1}\cdots w_n$, let $i$ be the smallest index such that $w_i\nu(R(v)). \end{equation} Returning to our example in Figure~\ref{fig:bicolored_tree_example}, we see that the internal vertices that are labeled by $1,2,3,4,5,6$ are all Lyndon since for each, their left child is a leaf. The internal vertex labeled as $7$ is also Lyndon but $8$ is not. To see why $7$ is Lyndon, note that $R(L(7))$ is the leaf labeled $9$ (and hence has valency 9) and $R(7)=5$ which has valency $2$. Thus, $ \nu(R(L(7)))>\nu(R(7))$. To see that $8$ is not Lyndon, note that $\nu(R(L(8)))=2\not> 3= \nu(R(8))$, i.e.~inequality~\ref{eq:1} is not satisfied. The Whitney duals for the weighted partition poset and the pointed partition poset can both be described using special types of normalized bicolored binary trees together with a sliding procedure used to merge such trees. We now discuss these special types of trees. \subsubsection{Pointed Lyndon forests} Let us define first the objects used to describe the Whitney dual of the pointed partition poset. \begin{defi} A normalized bicolored binary tree $T$ is said to be a \emph{pointed Lyndon tree} if for each internal vertex $v$ of $T$ such that $L(v)$ is not a leaf it must be that: \begin{equation}\label{condition:pointed_lyndon_1}\tag{PL1} \clr(L(v))\geq \clr(v) \end{equation} \begin{center} and \end{center} \begin{equation}\label{condition:pointed_lyndon_2}\tag{PL2} \text{If } \clr(L(v))=\clr(v)=\tre{1}\text{, then $v$ is a Lyndon node.} \end{equation} We say a forest $F$ is a \emph{pointed Lyndon forest} if all the connected components of $F$ are pointed Lyndon trees. We denote as $\Flyn_{n}^{\bullet}$ the set of all pointed Lyndon forests whose leaf label set is $[n]$. \end{defi} The tree in Figure~\ref{fig:bicolored_tree_example} is a pointed Lyndon tree. Indeed, we need only check the conditions for the internal vertices labeled by 7 and 8 since the left children of the other internal vertices are leaves. For 7, both it and its left child 6 are colored by \tre{1} (red) and 7 is Lyndon. For 8, it is colored \tbl{0} (blue) so we do not need to check anything else. \begin{figure} \begin{tikzpicture}[every node/.style={draw=black},every node/.append style={transform shape},scale=.8] \node[fill=red,circle](one){}; \node at ($(one)+(0.75,-1)$)[draw=none](two){2}; \node at ($(one)+(-0.75,-1)$)[scale=1.2,fill=blue](three){}; \node at ($(three)+(-0.75,-1)$)[draw=none](four){1}; \node at ($(three)+(0.75,-1)$)[draw=none](five){3}; \path[thick](one)edge(two)edge(three); \path[thick](three)edge(four)edge(five); \node at (0,-3) [draw=none] {\Large $T_1$}; \begin{scope}[shift={(5,0)}, every node/.style={draw=black},every node/.append style={transform shape}] \node[fill=blue,scale=1.2](one){}; \node at ($(one)+(0.75,-1)$)[draw=none](two){3}; \node at ($(one)+(-0.75,-1)$)[scale=1.2,fill=blue](three){}; \node at ($(three)+(-0.75,-1)$)[draw=none](four){1}; \node at ($(three)+(0.75,-1)$)[draw=none](five){$2$}; \path[thick](one)edge(two)edge(three); \path[thick](three)edge(four)edge(five); \node at (0,-3) [draw=none] {\Large $T_2$}; \end{scope} \node[fill=blue,scale=1.2,below of=two,align=center,anchor=center] at (1,-3)(label1){}; \node at ($(label1)+(1,0)$)[draw=none](label11){0}; \node[circle,fill=red,right of=label11,align=center,anchor=center](label2){}; \node at ($(label2)+(1,0)$)[draw=none](label22){1}; \end{tikzpicture} \caption{A bicolored Lyndon tree on the left (which is not a pointed Lyndon tree) and a pointed Lyndon tree on the right (which is not a bicolored Lyndon tree).}. \label{fig:bicoloredAndPointed} \end{figure} \subsubsection{Bicolored Lyndon forests} Let us now define the trees used for the Whitney dual of the weighted partition poset. \begin{defi} Let $T$ be a normalized bicolored binary tree, we say that $T$ is a \textit{bicolored Lyndon tree} if, for each internal vertex $v$ of $T$ whose left child is an internal vertex, either $v$ is a Lyndon vertex or it must be that: \begin{equation}\label{cleq:2}\tag{CL} \clr(L(v))>\clr(v). \end{equation} We say a forest $F$ is a \emph{bicolored Lyndon forest} if all the connected components of $F$ are bicolored Lyndon trees. We denote as $\Flyn_{n}^{w}$ the set of all bicolored Lyndon forests whose leaf label set is $[n]$. \end{defi} The tree in Figure~\ref{fig:bicolored_tree_example} is a bicolored Lyndon tree. As mentioned earlier, the only internal vertex that is not a Lyndon vertex is the one labeled by 8. Its color is \tbl{0} (blue) and its left child is colored by \tre{1} (red), so it satisfies condition~\eqref{cleq:2}. At this point, the reader may be wondering if pointed Lyndon trees and bicolored Lyndon trees are the same. To see that this is not the case, consider the trees in Figure~\ref{fig:bicoloredAndPointed}. We claim that the tree $T_1$ is bicolored, but not pointed. All the internal vertices of $T_1$ are Lyndon, so it is automatically a bicolored Lyndon tree. However, it is not a pointed Lyndon tree because the color of the root is larger than its left child, a violation of condition~\eqref{condition:pointed_lyndon_1}. On the other hand, we claim that $T_2$ is a pointed Lyndon tree, but not a bicolored Lyndon tree. It is pointed since both vertices are colored by \tbl{0} (blue). However, it is not a bicolored Lyndon tree since the root is not a Lyndon vertex and the root's color is not strictly larger than its left child, a violation of condition~\eqref{cleq:2}. \begin{rema} Note that condition ~\eqref{cleq:2} implies that the family of classical Lyndon trees coincide with the subfamily of bicolored Lyndon trees that either have only internal vertices of color \tbl{0} (blue) or that have only internal vertices of color \tre{1} (red). On the other hand condition ~\eqref{condition:pointed_lyndon_2} implies that the family of classical Lyndon trees coincide with the subfamily of pointed Lyndon trees which have all internal nodes colored \tre{1} (red). \end{rema} \subsection{A Whitney dual for \texorpdfstring{$\Pi_n^\bullet$}-} \label{subsection:whitney_dual_pointed} Let $F$ be a pointed Lyndon forest. We explain how to associate an ascent-free saturated chain $c(F)$ starting at $\hat{0}$ in $\Pi_n^\bullet$. Recall that $F$ has a unique reverse-minimal linear extension order on the internal vertices. To construct an ascent-free chain from $F$, we start with the bottom element, $\tilde{1}/\tilde{2}/\cdots /\tilde{n}$. In the first step, we merge together the two blocks which are in the left subtree and right subtree rooted at the first internal vertex. We keep the point on the element of the left tree if the first internal vertex is colored \tre{1} (red) and we keep the point on the element of the right tree if it is colored is \tbl{0} (blue). We continue doing this so that at the $i^{th}$ step we merge together the elements of the left and right subtrees rooted at the $i^{th}$ internal vertex and keep the pointed element of the left subtree if it is colored~\tre{1} (red) or the right subtree if it is colored \tbl{0} (blue). Figure~\ref{fig:pointed_Lyndon_Tree_and_chain} has a depiction of a pointed Lyndon forest $F$ and its corresponding chain $c(F)$. In the first step of the chain, we merge the blocks containing $6$ and $7$ since they are the leafs in the left and right tree rooted at the first internal vertex. We keep $6$ pointed because this first internal vertex is colored by \tre{1} (red). Continuing this process gives the saturated chain seen in the figure. Let $F$ be a pointed Lyndon forest and let $a_i$ be the valency of left child of the $i^{th}$ internal vertex and $b_i$ the valency of right child of $i^{th}$ internal vertex. That is $a_i = \nu(L(i))$ and $b_i =\nu(R(i))$. Note that $a_i$ is also the valency of $i$ since our trees are normalized (i.e.~$\nu(i) =\nu(L(i)))$. Then, it is straightforward to see that the sequence of labels along the chain $c(F)$ is $(a_1,b_1)^{u_1}(a_2,b_2)^{u_2}\cdots (a_{k},b_{k})^{u_{k}}$ where $u_i$ is the color of the $i^{th}$ internal vertex. Let us return to the forest in Figure~\ref{fig:pointed_Lyndon_Tree_and_chain}. Since the left child of the first internal vertex is the leaf labeled 6, the right child is a leaf labeled 7, and the first internal vertex is colored \tre{1} (red), we have that $(a_1,b_1)^{u_1} = (6,7)^1$. Moving to the second internal vertex we see its left child is the leaf $5$ and the right child is the leaf $8$. Since it is colored by \tre{1} (red) the next label is $(5,8)^{\tre{1}}$. Next, moving to the third internal vertex, we see that the left child is a leaf $4$ and the right child is the first internal vertex whose valency is 6. Since the third internal vertex is colored \tre{1} (red), the corresponding label is $(4,6)^{\tre{1}}$. Continuing this, we see the sequence we get from the forest is $$ (6,7)^{\tre{1}} (5,8)^{\tre{1}} (4,6)^{\tre{1}} (3,4)^{\tbl{0}} (2,5)^{\tbl{0}} (1,9)^{\tre{1}} (1,2)^{\tre{1}} $$ Note that the sequence (and hence its chain) is ascent-free with respect to the ordering of the labels for $\lambda_\bullet$. It turns out that this not a coincidence as we show next. \begin{figure} \centering \resizebox{\textwidth}{!}{ \begin{tikzpicture}[scale=0.8] \node[](one){}; \node[] at (0,-6) {$F$}; \node at ($(one)+(-1.5,-1.5)$)(two){}; \tikzstyle{every node}=[draw,scale=0.8] \node at ($(one)+(-1.5,-1.5)$)[circle,red](two){7}; \node at ($(two)+(-1,-1)$)[circle,red](three){6}; \node at ($(three)+(-0.75,-1)$)[draw=none](3son1){1}; \node at ($(three)+(0.75,-1)$)[draw=none](3son2){9}; \node at ($(two)+(1,-1)$)[blue,scale=1.2](four){5}; \node at ($(four)+(-0.75,-1)$)[draw=none](4son){2}; \node at ($(four)+(0.75,-1)$)[circle,red](4sonn){2}; \node at ($(one)+(1.5,-1.5)$)[blue,scale =1.2](five){4}; \node at ($(five)+(0.75,-1)$)[circle,red](six){3}; \node at ($(six)+(-0.75,-1)$)[draw=none](6son){4}; \node at ($(4sonn)+(-0.75,-1)$)[draw=none](4sonn1){5}; \node at ($(4sonn)+(0.75,-1)$)[draw=none](4sonn2){8}; \node at ($(five)+(-0.75,-1)$)[draw=none](5son1){3}; \node at ($(six)+(0.75,-1)$)[circle,red](6sonn){1}; \node at ($(6sonn)+(-0.75,-1)$)[draw=none](last1){6}; \node at ($(6sonn)+(0.75,-1)$)[draw=none](last2){7}; \path[thick](six)edge(6son)edge(6sonn); \path[thick](6sonn)edge(last1)edge(last2); \path[thick](five)edge(5son1); \path[thick](4sonn)edge(4sonn1)edge(4sonn2); \path[thick](two)edge(three)edge(four); \path[thick](five)edge(six); \path[thick](three)edge(3son1)edge(3son2); \path[thick](four)edge(4son)edge(4sonn); \node[fill=blue,scale=1.2,below of=two,align=center,anchor=center,xshift=0.25cm,yshift=-2cm](label1){}; \node at ($(label1)+(1,0)$)[draw=none](label11){0}; \node[circle,fill=red,right of=label11,align=center,anchor=center](label2){}; \node at ($(label2)+(1,0)$)[draw=none](label22){1}; \begin{scope} [shift = {(10, -5)}] \tikzstyle{every node}=[] \node[] at (0,-1) {$c(F)$}; \node (0) at (0,0) {$\tilde{1}/\tilde{2}/\tilde{3}/\tilde{4}/\tilde{5}/\tilde{6}/\tilde{7}/\tilde{8}/\tilde{9}$}; \node (1) at (0,1.25) {$\tilde{1}/\tilde{2}/\tilde{3}/\tilde{4}/\tilde{5}/\tilde{6} 7/\tilde{8}/\tilde{9}$}; \draw (0)--(1); \node (2) at (0,2.5) {$\tilde{1}/\tilde{2}/\tilde{3}/\tilde{4}/\tilde{5}8/\tilde{6}7/\tilde{9}$}; \draw (1)--(2); \node (3) at (0,3.75) {$\tilde{1}/\tilde{2}/\tilde{3}/\tilde{4}67 /\tilde{5}8/\tilde{9}$}; \draw (2)--(3); \node (4) at (0,5) {$\tilde{1}/\tilde{2}/3\tilde{4}67 /\tilde{5}8/\tilde{9}$}; \draw (3)--(4); \node (5) at (0,6.25) {$\tilde{1}/2\tilde{5}8/3\tilde{4}67/\tilde{9}$}; \draw (4)--(5); \node (6) at (0,7.5) {$ \tilde{1}9/2\tilde{5}8/3\tilde{4}67 $}; \draw (5)--(6); \node (7) at (0,8.75) {$ \tilde{1}2589/3\tilde{4}67 $}; \draw (6)--(7); \end{scope} \end{tikzpicture} } \caption{Pointed Lyndon forest $F$ and corresponding ascent-free chain $c(F)$.}\label{fig:pointed_Lyndon_Tree_and_chain} \end{figure} \begin{theo}\label{thm:bijectionBetweenForestAndChain} The map sending $F$ to $c(F)$ is a bijection between pointed Lyndon forest whose leaf label set is $[n]$ and ascent-free chains starting at $\hat{0}$ of $\Pi_n^\bullet$, where the ascent-free condition is defined with respect to $\lambda_\bullet$. \end{theo} \begin{proof} First, we show that the map is well-defined. That is, $c(F)$ is in fact ascent-free for all pointed Lyndon forest $F$. Assume that the internal vertices are $1,2,\dots, k$ which is also the reverse-minimal ordering. Let $(a_1,b_1)^{u_1}(a_2,b_2)^{u_2}\cdots (a_{k},b_{k})^{u_{k}}$ be the sequence of labels along $c(F)$. As mentioned earlier, $a_i$ is the valency of $i$. Since we are using the reverse-minimal order, this gives us $a_1\geq a_2\geq \cdots \geq a_k$. If $a_i> a_{i+1}$, then $(a_i,b_i)^{u_i} > (a_{i+1},b_{i+1})^{u_{i+1}}$ in the ordering of labels of $\lambda_\bullet$. On the other hand, if $a_i=a_{i+1}$, then $i+1$ is an ancestor of $i$ and $i$ must be in the left tree rooted at $i+1$. Since the reverse-minimal ordering is a linear extension of the internal vertices, it must be the case that $i$ is the left child of $i+1$. Since $F$ is a pointed Lyndon forest, condition~\eqref{condition:pointed_lyndon_1} implies that $u_i = \clr(i) \geq \clr(i+1) =u_{i+1}$. If $i+1$ is not Lyndon, then condition~\eqref{condition:pointed_lyndon_2} implies that $u_i>u_{i+1}$. Since $u_i,u_{i+1}\in \{\tbl{0},\tre{1}\}$, this implies that $u_i=\tre{1}$ and $u_{i+1}=\tbl{0}$. So, if $i+1$ is not Lyndon, $(a_i,b_i)^{u_i} > (a_{i+1},b_{i+1})^{u_{i+1}}$. On the other hand, if $i+1$ is Lyndon, $b_{i+1}=\nu(R(i+1)) <\nu(R(L(i+1))) =b_{i}$. Then either $(a_i,b_i)^{u_i} > (a_{i+1},b_{i+1})^{u_{i+1}}$ (if $u_i>u_{i+1}$ or $u_i=1=u_{i+1}$) or $(a_i,b_i)^{u_i}$ and $(a_{i+1},b_{i+1})^{u_{i+1}}$ are incomparable (if $u_i=\tbl{0}=u_{i+1})$. It follows that the map is well-defined. We claim that the map sending $F$ to $c(F)$ is invertible. Suppose we have an ascent-free chain with sequence $$ (a_1,b_1)^{u_1}(a_2,b_2)^{u_2}\cdots (a_{k},b_{k})^{u_{k}}. $$ Build a forest recursively by first placing $n$ isolated vertices (which will be the leafs) labeled by $[n]$. Now assume that you are at the $i^{th}$ step of this process. Let~$T_1$ be the connected component of the forest with minimal leaf label $a_i$ and let $T_2$ be the connected component of the forest with minimal leaf label $b_i$. Add a vertex colored $u_i$ and add edges from this vertex to the roots of~$T_1$ and $T_2$. Repeat this process until each pair $(a_j,b_j)^{u_j}$ has been used and call the resulting forest $F$. Since $ (a_1,b_1)^{u_1}(a_2,b_2)^{u_2}\cdots (a_{k},b_{k})^{u_{k}}$ is ascent-free, it must be the case that $a_1\geq a_2\geq \cdots \geq a_k$. It then follows that the reverse minimal ordering on $F$ is exactly the order that the internal vertices were added in the process. Upon observing this, it is clear that $c(F)$ is the chain with label sequence $(a_1,b_1)^{u_1}(a_2,b_2)^{u_2}\cdots (a_{k},b_{k})^{u_{k}}$. So, if we can show this map is well-defined, we will have that the map sending $F$ to $c(F)$ is invertible and thus a bijection. We do this next. Let $F$ be a forest obtained by using the inverse procedure described in the previous paragraph. We need to show that $F$ is a pointed Lyndon forest. First, it is clear from the construction that $F$ is a bicolored binary tree. Since $a_ib_i$ (otherwise either a label would be repeated in the sequence or the sequence would have an ascent). Thus, $\nu(R(L(v))= b_{i-1} > b_i = \nu(R(v))$ and so $v$ is a Lyndon vertex (equation~\eqref{eq:1}). It follows that condition~\eqref{condition:pointed_lyndon_2} is also satisfied. We conclude that~$F$ is a pointed Lyndon forest. Thus, the inverse map is well-defined, completing the proof. \end{proof} By the previous theorem and Theorem~\ref{thm:WhitneyDualRLambda}, we can describe the Whitney dual of the pointed partition poset using pointed Lyndon forests. To do this, we will need to describe how to merge trees in a Lyndon forest. Suppose that $T_1$ and $T_2$ are trees in a Lyndon forest with roots $r_1$ and $r_2$ where the minimum leaf label of $T_1$ is less than the minimum leaf label of $T_2$. Let $u\in \{\tbl{0},\tre{1}\}$. To \emph{$u$-merge} $T_1$ with $T_2$, we first create a new vertex $r$ and color it so $\clr(r)=u$. Then we add edges from $r$ to $r_1$ and $r_2$. If the resulting tree is a pointed Lyndon tree, we stop. If it is not, we \emph{slide} the new internal vertex $r$ together with its right subtree past its left child and check if the result is a pointed Lyndon tree. We continue this process until we obtain a pointed Lyndon tree. \begin{figure} \centering \resizebox{\textwidth}{!}{ \begin{tikzpicture}[scale=0.8] \tikzstyle{every node} = [draw=black] \node[fill=blue,scale=1.2](root){}; \node at ($(root)+(1.1,-1)$)[inner sep=3pt,fill=red,circle](one){}; \node at ($(one)+(-0.75,-1)$)[draw=none](two){2}; \node at ($(one)+(0.75,-1)$)[draw=none](three){3}; \path[thick](one)edge(two)edge(three); \node at ($(root)+(-1.1,-1)$)[inner sep=3pt,fill=red,circle](one1){}; \node at ($(one1)+(-0.75,-1)$)[draw=none](two1){1}; \node at ($(one1)+(0.75,-1)$)[draw=none](three1){4}; \path[thick](one1)edge(two1)edge(three1); \path[thick](root)edge(one1)edge(one); \node at ($(root)+(0,-2.6)$)[draw=none](arrow1){$T_1$}; \node at ($(root)+(4,0)$)[scale=1.2,fill=blue](onee){}; \node at ($(onee)+(0.75,-1)$)[draw=none](twoo){6}; \node at ($(onee)+(-0.75,-1)$)[inner sep=3pt,fill=red,circle](threee){}; \node at ($(threee)+(-0.75,-1)$)[draw=none](fourr){5}; \node at ($(threee)+(0.75,-1)$)[draw=none](fivee){7}; \path[thick](onee)edge(twoo)edge(threee); \path[thick](threee)edge(fourr)edge(fivee); \node at ($(onee)+(0,-2.6)$)[draw=none](arrow1){$T_2$}; \node at ($(one)+(0.7,-2.4)$)[draw=none](o){}; \node at ($(o)+(5,2)$)[draw=none](arrow1){$\longrightarrow$}; \node at ($(o)+(10,3.5)$)[inner sep=3pt,red,circle](rootp){r}; \node at ($(o)+(8,2.5)$)[fill=blue,scale=1.2](root1){}; \node at ($(root1)+(1.1,-1)$)[inner sep=3pt,fill=red,circle](one1){}; \node at ($(one1)+(-0.75,-1)$)[draw=none](two1){2}; \node at ($(one1)+(0.75,-1)$)[draw=none](three1){3}; \path[thick](one1)edge(two1)edge(three1); \node at ($(root1)+(-1.1,-1)$)[inner sep=3pt,fill=red,circle](one11){}; \node at ($(one11)+(-0.75,-1)$)[draw=none](two11){1}; \node at ($(one11)+(0.75,-1)$)[draw=none](three11){4}; \path[thick](one11)edge(two11)edge(three11); \path[thick](root1)edge(one11); \path[densely dotted](root1)edge(one1); \node at ($(root1)+(4,0)$)[scale=1.2,fill=blue](onee1){}; \node at ($(onee1)+(0.75,-1)$)[draw=none](twoo1){6}; \node at ($(onee1)+(-0.75,-1)$)[inner sep=3pt,fill=red,circle](threee1){}; \node at ($(threee1)+(-0.75,-1)$)[draw=none](fourr1){5}; \node at ($(threee1)+(0.75,-1)$)[draw=none](fivee1){7}; \path[thick](onee1)edge(twoo1)edge(threee1); \path[thick](threee1)edge(fourr1)edge(fivee1); \path[densely dotted](rootp)edge(onee1); \path[thick](rootp)edge(root1); \node at ($(o)+(10,-1)$)[draw=none](arrow1){$\downarrow$}; \node at ($(arrow1)+(0,-1)$)[fill=blue,scale=1.2](rootp2){}; \node at ($(arrow1)+(-2,-2.5)$)[inner sep=3pt,red,circle](root12){r}; \node at ($(root12)+(4,0)$)[inner sep=3pt,fill=red,circle](one12){}; \node at ($(one12)+(-0.75,-1)$)[draw=none](two12){2}; \node at ($(one12)+(0.75,-1)$)[draw=none](three12){3}; \path[thick](one12)edge(two12)edge(three12); \node at ($(root12)+(-1.1,-1)$)[inner sep=3pt,fill=red,circle](one112){}; \node at ($(one112)+(-0.75,-1)$)[draw=none](two112){1}; \node at ($(one112)+(0.75,-1)$)[draw=none](three112){4}; \path[thick](one112)edge(two112); \path[densely dotted](one112)edge(three112); \node at ($(root12)+(1.1,-1)$)[scale=1.2,fill=blue](onee12){}; \node at ($(onee12)+(0.75,-1)$)[draw=none](twoo12){6}; \node at ($(onee12)+(-0.75,-1)$)[inner sep=3pt,fill=red,circle](threee12){}; \path[densely dotted](root12)edge(onee12); \path[thick](root12)edge(one112); \node at ($(threee12)+(-0.75,-1)$)[draw=none](fourr12){5}; \node at ($(threee12)+(0.75,-1)$)[draw=none](fivee12){7}; \path[thick](onee12)edge(twoo12)edge(threee12); \path[thick](threee12)edge(fourr12)edge(fivee12); \path[thick](rootp2)edge(one12); \path[thick](rootp2)edge(root12); \filldraw[black] (16,-8) node (5.6,-14.15)[anchor=west][draw=none,below](e12) {}; \node at ($(e12)+(0,-1)$)[draw=none](arrow1){}; \node at ($(o)+(5,-3)$)[draw=none](arrow1){$\longleftarrow$}; \node at ($(o)+(0,-2)$)[fill=blue,scale=1.2](froot){}; \node at ($(o)+(-2,-3.5)$)[inner sep=3pt,fill=red,circle](froot1){}; \node at ($(froot1)+(4,0)$)[inner sep=3pt,fill=red,circle](fone1){}; \node at ($(fone1)+(-0.75,-1)$)[draw=none](ftwo1){2}; \node at ($(fone1)+(0.75,-1)$)[draw=none](fthree1){3}; \path[thick](fone1)edge(ftwo1)edge(fthree1); \node at ($(froot1)+(1.1,-1)$)[draw=none](fone11){4}; \node at ($(froot1)+(-1.1,-1)$)[inner sep=3pt,red,circle](f1){r}; \node at ($(f1)+(-.75,-1)$)[draw=none](f11){1}; \node at ($(f1)+(.75,-1)$)[scale=1.2,fill=blue](f12){}; \node at ($(f12)+(-.75,-1)$)[inner sep=3pt,fill=red,circle](f21){}; \node at ($(f21)+(-.75,-1)$)[draw=none](f31){5}; \node at ($(f21)+(.75,-1)$)[draw=none](f32){7}; \node at ($(f12)+(.75,-1)$)[draw=none](f22){6}; \path[thick](froot)edge(froot1)edge(fone1); \path[thick](froot1)edge(fone11)edge(f1); \path[thick](f1)edge(f11)edge(f12); \path[thick](f12)edge(f21)edge(f22); \path[thick](f21)edge(f31)edge(f32); \end{tikzpicture} } \caption{Pointed Lyndon tree obtained by a $1$-merge of two pointed Lyndon trees $T_1$ and $T_2$.} \label{fig:my_2-merge-pointed} \end{figure} An example of a $1$-merge of two pointed Lyndon trees is illustrated in Figure~\ref{fig:my_2-merge-pointed}. First, we add a vertex $r$ colored by \tre{1} (red) and add edges from $r$ to the roots of~$T_1$ and $T_2$. We then need to check whether or not this construction results into a pointed Lyndon tree. Since $\clr(L(r))=\tbl{0}<\tre{1}=\clr(r)$, we have a violation of condition~\eqref{condition:pointed_lyndon_1} at $r$ and hence, this is not yet a pointed Lyndon tree. We then slide $r$ together with its right subtree, to its left, interchanging $r$ and $L(r)$. After sliding $r$ to its left we have that $\clr(L(r))=\clr(r)=\tre{1}$. However, $4=v(R(L(r)))