% Author: Samuele Giraudo, Yannic Vargas
% Creation: sep. 2023
% Modifications: sep. 2023, oct. 2023, nov. 2023, feb. 2024, may 2024, jun. 2024, nov. 2024,
% 2025-03

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\title{Operads of packed words, quotients, and monomial bases}
%
\author{\texorpdfstring{%
    Samuele Giraudo%
    \thanks{
        \href{mailto:giraudo.samuele@uqam.ca}{giraudo.samuele@uqam.ca}.
         This research has been partially supported by the Natural Sciences and Engineering
        Research Council of Canada (RGPIN-2024-04465).
        }%
    \addressmark{1}
    \and
    Yannic Vargas%
    \thanks{
        \href{mailto:yannic.vargas@cunef.edu}{yannic.vargas@cunef.edu}.
        }%
    \addressmark{2}
    }{%
    Samuele Giraudo%
    %
    and
    Yannic Vargas%
    %
    }}
%
\address{%
    \addressmark{1} Université du Québec à Montréal, LACIM, Pavillon Président-Kennedy, 201
    Avenue du Président-Kennedy, Montréal, H2X~3Y7, Canada. \\
    \addressmark{2} CUNEF University, Department of Mathematics, C. de Almansa, 101, 28040
    Madrid, Spain.}
%
\abstract{%
    The associative operad is a central structure in operad theory, defined on the linear
    span of permutations. We build two analogs of the associative operad on the linear span
    of packed words. By seeing a packed word as a surjective map between two finite sets,
    our first operad is graded by the cardinality of the domain and the second one, by the
    cardinality of the codomain. In the same way as the associative operad of permutations
    admits as quotients the duplicial and interstice operads, we derive similar structures
    for our operads of packed words. Both operadic operations on monomial bases, constructed
    from partial orders on packed words, are monomial-positive. We propose also an analogue
    of Dynkin idempotent of Zie algebras in this context of operads of packed words.
}
%
\keywords{%
    Operad;
    Permutation;
    Packed word;
    Dynkin idempotent.}
%
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\begin{document}
\maketitle

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\section*{Introduction}
The associative operad $\As$ is an algebraic structure playing a central role in the operad
theory. It is defined on the linear span of permutations and its partial composition
consists of inserting a permutation into another one, interpreted as permutation matrices.
The first reason justifying the importance of $\As$ is that it intervenes in a crucial way
in the description of the axioms of symmetric operads. A second reason, shown by Aguiar and
Livernet~\cite{AL2007}, relates to its richness from a combinatorial point of view. Indeed,
$\As$ admits a basis (called \emph{monomial basis}) defined using the left weak order on
permutations which has the nice property that the partial composition of two elements of
this basis is a sum of an interval of this partial order. Furthermore, this operad enjoys
many properties since it admits not only the Lie operad as a suboperad, but also, as
quotients, the duplicial operad of binary trees~\cite{BF03} and the interstice operad on two
generators of binary words~\cite{Chapoton2002,CG22}.

These three operads on permutations, binary trees, and binary words form a hierarchy very
similar to a well-known hierarchy of combinatorial Hopf algebras involving the same spaces
of combinatorial objects, namely, the Malvenuto--Reutenauer Hopf algebra of
permutations~\cite{MR1995,DHT02}, the Loday--Ronco Hopf algebra of binary
trees~\cite{LR1998}, and the noncommutative symmetric functions Hopf
algebra~\cite{GKLLRT94}. Interestingly enough, there is a generalization of the
Malvenuto--Reutenauer Hopf algebra on the linear span of packed words.  This  Hopf algebra
has been introduced by Hivert~\cite{Hiv99} when he considered a notion of word
quasi-symmetric functions. This construction is natural since while permutations are
bijections, packed words are surjections. In this context, the analog of the Loday--Ronco
Hopf algebra involves Schröder trees~\cite{NT20} and the analog of the noncommutative
symmetric functions  Hopf algebra involves ternary words~\cite{NT20}. The starting point of
the present work is to explore whether such a natural generalization of $\As$ exists and if
it leads to a similar hierarchy of operads.

Our main contribution consists of the introduction of two different generalizations of $\As$
on the linear span of packed words. By seeing a packed word as its matrix, we obtain a right
version $\RightPAs$ consisting of inserting a packed word matrix at a given position into
another one, and a left version $\LeftPAs$ consisting of inserting several copies of a
packed word matrix onto given values. These operads also differ in the way the arity of a
packed word is defined: in $\RightPAs$ (resp.\ $\LeftPAs$), the arity of a packed word is
the cardinality of its domain (resp.\ codomain). As they are not isomorphic as graded
spaces, $\RightPAs$ and $\LeftPAs$ are not isomorphic as operads. Moreover, $\LeftPAs$ is a
symmetric operad while $\RightPAs$ is not. Besides, the operad $\LeftPAs$ is not
combinatorial in the sense that it admits infinitely many elements of any given arity $n
\geq 1$. Nevertheless, despite this fact, this operad is rich from a combinatorial point of
view since it admits several quotients using well-known equivalence relations on packed
words, like the sylvester~\cite{HNT05}, hypoplactic~\cite{KT97}, or Baxter~\cite{Gir12}
congruences.

This work is presented as follows. \Cref{sec:preliminaries} contains fundamental
notions about the main combinatorial objects and operads appearing in this work. In
\Cref{sec:operads_packed_words}, we introduce and construct the operads $\RightPAs$
and $\LeftPAs$ and present their first properties. \Cref{sec:quotients} is devoted to
the study of some of the quotients of these two operads of packed words which involve other
families of combinatorial objects. After recalling the definition of \emph{left and right
weak order for packed words}, we use these orders in \Cref{sec:dynkin} to construct
monomial bases and to study their behavior under our two operads. An analogue identity of
the classical Dynkin idempotents for \emph{Zie algebras}~\cite{AM2017} in term of monomials
is derived.

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{\it General notations and conventions.}
For any integer $i$, $[i]$ denotes the set $\{1, \dots, i\}$. For any set $A$, $A^*$ is the
set of words on $A$. For any $w \in A^*$, $\Length(w)$ is the length of $w$, and for any $i
\in [\Length(w)]$, $w(i)$ is the $i$-th letter of $w$. The only word of length $0$ is the
empty word $\epsilon$. For any $1 \leq i \leq j \leq \Length(w)$, $w(i, j)$ is the word
$w(i) \dots w(j)$. Given two words $w$ and $w'$, the  concatenation of $w$ and $w'$ is
denoted by $ww'$ or by~$w \Conc w'$.

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\section{Preliminaries} \label{sec:preliminaries}

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\subsection{Packed words}
Let $\PositiveN$ be the set $\N \setminus \{0\}$. For any $u \in \PositiveN^*$ and $\alpha,
\beta \in \N$, let $\Increment{\alpha}{\beta}(u)$ be the word obtained by incrementing by
$\alpha$ the letters of $u$ which are greater than $\beta$. For instance,
$\Increment{2}{4}(124\ColB{5}4\ColB{6}) = 124\ColB{7}4\ColB{8}$.  For any $u, v \in
\PositiveN^*$ and $i \in \PositiveN$, let $u \Replace_i v$ be the word on $\PositiveN^*$
obtained by replacing each occurrence of $i$ in $u$ by a copy of $v$. For instance
$\ColB{2}1\ColB{2}3 \Replace_{\ColB{2}} \ColB{41} = \ColB{41} 1 \ColB{41} 3$ and $21
\Replace_{\ColB{3}} \ColB{311} = 21$. Let $w \in \PositiveN^*$. The \Def{alphabet} of $w$ is
the set $\text{Alph}(w)$ formed by the letters appearing in $w$. An \Def{inversion} of $w$
is a pair $(i,j)$ such that $1 \leq i < j \leq \Length(w)$ and $w(i)>w(j)$. The \Def{set of
inversions} of $w$ is denoted by $\InversionSet(w)$ and the \Def{number of inversions} of
$w$ is $\Inversions(w):=|\InversionSet(w)|$. The \Def{standardization map}~\cite{Hivert2006}
$\st: \PositiveN^n \to \mathfrak{S}_n$ is the function sending $w$ to the unique permutation
$\st(w) \in \mathfrak{S}_{\Length(w)}$ such that $\InversionSet(w) = \InversionSet(\st(w))$.
For instance, if $x<y<z$ are letters of $\PositiveN$, then $\text{st}(yyxzzxzyx)=451782963
\in \mathfrak{S}_9$.

A \Def{packed word} is a word $w \in \PositiveN^*$ satisfying $i - 1 \in \text{Alph}(w)$
whenever $i \in \text{Alph}(w)$ and $i \geq 2$. This implies that $\text{Alph}(w)=[k]$ for
some $k \geq 0$ and that $k \leq \Length(w)$. For any $k,n \geq 0$, let $\mathfrak{P}_n[k]$
be the set of all packed words of alphabet $[k]$ and length $n$.  When $n = k$, this set
coincides with $\SetPermutations_n$.

Recall that the symmetric group acts on the vector space of non-commutative polynomials
$\mathbb{K}\langle \mathsf{X}\rangle$, for any countable set $\mathsf{X}$, where
$\mathbb{K}$ is any field of characteristic zero. When $\mathsf{X}=\PositiveN$, this action
restricts to packed words: if $w \in \mathfrak{P}_n[k]$ and $\sigma \in \mathfrak{S}_n$,
then
\begin{equation} \label{equ:right_action_permutation}
    w \cdot \sigma:= w(\sigma(1))\cdots w(\sigma(n)) \in \mathfrak{P}_n[k].
\end{equation}
Indeed, if $w$ is a packed word, any permutation of its letters is also a packed word.


For any $w \in \mathfrak{P}_n[k]$,  the \Def{composition type} $\chi(w)$ of $w$ is the
unique weakly increasing word obtained after rearrangement of $w$. In particular, $\chi(w)
\in \mathfrak{P}_n[k]$. Every weakly increasing packed word $w \in \mathfrak{P}_n[k]$ can be
encoded by the integer composition $c_1c_2 \dots c_k$ of $n$ such that every $i$ appears
$c_i$ times in $w$. Equivalently, through the classical bijection between subsets of $[n-1]$
and binary strings of length $n-1$, the composition type $\chi(w)$ of a packed word of
length $n$ can be also described as a word of length $n-1$ on the alphabet $\{1,2\}$. For
instance,
\begin{math}
    \chi(2311223) = 1122233 \leftrightarrow \ColA{-}\ColB{+}\ColA{--}\ColB{+}\ColA{-}
    \leftrightarrow \ColA{1}\ColB{2}\ColA{11}\ColB{2}\ColA{1}
\end{math}
and
\begin{math}
    \chi(221) = 122  \leftrightarrow \ColB{+}\ColA{-} \leftrightarrow \ColB{2}\ColA{1}.
\end{math}

The following lemma, due to Hivert, relates a packed word with its composition type.

\begin{Lemma}[Hivert, \cite{Hivert2006}]\label{stHiv2}
     For any $w \in \mathfrak{P}_n[k]$,
     \begin{math}
         w = \chi(w)\cdot \text{st}(w).
     \end{math}
    Moreover, $\text{st}(w)$ is the smallest permutation in $\mathfrak{S}_n$ for the right
    weak order satisfying the above property.
\end{Lemma}

This result implies that every packed word $w$ is encoded by its composition type and its
standardization: the composition type tells us how many times each letter appears in $w$,
while $\text{st}(w)$ encodes the relative order of the appearance of the letters in~$w$.

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\subsection{Associative operad and quotients}
We follow the usual notations about symmetric unital operads~\cite{LV2012} (called simply
\Def{operads} here). Here are four important examples of (symmetric or nonsymmetric) operads.
\begin{itemize}[fullwidth]
    \item The \Def{right associative operad} is the symmetric operad $\RightAs$ (also
    written $\As$) wherein for any $n \geq 1$, $\RightAs(n)$ is the linear span of the set
    $\mathfrak{S}_n$. The set $\{\BasisE_\sigma : \sigma \in \mathfrak{S}_n, \, n \geq 1\}$
    is a basis of $\RightAs$. Given $\alpha \in \mathfrak{S}_n$, $\beta \in \mathfrak{S}_m$
    and $i \in [n]$, let
    \begin{equation}\label{defB}
        B_i(\alpha, \beta) :=
            \Increment{m - 1}{\alpha(i)}(\alpha(1, i - 1))
            \; \Conc \;
            \Increment{\alpha(i) - 1}{0}(\beta)
            \; \Conc \;
            \Increment{m - 1}{\alpha(i) - 1}(\alpha(i + 1, n)).
    \end{equation}
    For instance,
    \begin{math}
        B_5(\ColA{3612} \ColB{4} \ColA{57}, \ColB{231}) = \ColA{3812} \ColB{564} \ColA{79}.
    \end{math}
    The \ColB{red} labels reflect the \ColB{red} permutation $231$ inserted into the
    \ColA{blue} permutation $3612457$ onto the letter $\ColB{4}$ at the $5$-th position. The
    permutation $B_i(\alpha, \beta)$ is sometimes called the \Def{block permutation}
    associated to $\alpha$ and $\beta$ (see~\cite{LV2012}). The partial composition of
    $\RightAs$ satisfies $\BasisE_\sigma \circ_i \BasisE_\nu = \BasisE_{B_i(\sigma, \nu)}$
    and the action of the symmetric group $\mathfrak{S}_n$ satisfies $\BasisE_\sigma \cdot
    \nu = \BasisE_{\sigma \circ \nu}$ where $\circ$ is the composition of permutations.
    %
    \item The \Def{left associative operad} is the symmetric operad $\LeftAs$ defined on the
    same space as the one of $\RightAs$, seen on the same $\BasisE$-basis. The partial
    composition of $\LeftPAs$ satisfies $\BasisE_\sigma \circ_i \BasisE_\nu = \BasisE_\pi$
    where $\pi$ is the permutation obtained by replacing in $\Increment{m - 1}{i}(\sigma)$
    the letter $i$ by $\Increment{i - 1}{0}(\nu)$, where $m$ is the greatest letter of
    $\nu$. The action of the symmetric group $\mathfrak{S}_n$ satisfies $\BasisE_\sigma
    \cdot \nu = \BasisE_{\nu^{-1} \circ \sigma}$ where $\circ$ is the composition of
    permutations. The operads $\RightAs$ and $\LeftAs$ are isomorphic through the linear map
    $\phi : \RightAs \to \LeftAs$ satisfying $\phi\left(\BasisE_\sigma\right) =
    \BasisE_{\sigma^{-1}}$.
    %
    \item The \Def{duplicial operad}~\cite{BF03} is the nonsymmetric operad $\Dup$ wherein
    for any $n \geq 1$, $\Dup(n)$ is the linear span of the set $\mathfrak{B}_n$ of binary
    trees with $n$ internal nodes. The set $\{\BasisE_\Tree : \Tree \in \mathfrak{B}_n, \, n
    \geq 1\}$ is a basis of $\Dup$. The partial composition of $\Dup$ satisfies
    $\BasisE_\Tree \circ_i \BasisE_\TreeS = \BasisE_\TreeR$ where $\TreeR$ is the binary
    tree obtained by replacing the $i$-th internal node $u$ of $\Tree$ for the infix
    traversal by a copy of $\TreeS$ and by grafting the left subtree of $u$ on the leftmost
    leaf of the copy and by grafting the right subtree of $u$ on the rightmost least of the
    copy. For instance, in $\Dup$, we have
    \begin{equation}
        \BasisE_{
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                \node[Leaf](14)at(14.00,-9.00){};
                \node[Leaf](2)at(2.00,-9.00){};
                \node[Leaf](4)at(4.00,-6.00){};
                \node[Leaf](6)at(6.00,-9.00){};
                \node[Leaf](8)at(8.00,-12.00){};
                \node[Node](1)at(1.00,-6.00){};
                \node[Node,draw=ColB,fill=ColB!50](11)at(11.00,-3.00){};
                \node[Node](13)at(13.00,-6.00){};
                \node[Node](3)at(3.00,-3.00){};
                \node[Node](5)at(5.00,0.00){};
                \node[Node](7)at(7.00,-6.00){};
                \node[Node](9)at(9.00,-9.00){};
                \draw[Edge](0)--(1);
                \draw[Edge](1)--(3);
                \draw[Edge](10)--(9);
                \draw[Edge](11)--(5);
                \draw[Edge](12)--(13);
                \draw[Edge](13)--(11);
                \draw[Edge](14)--(13);
                \draw[Edge](2)--(1);
                \draw[Edge](3)--(5);
                \draw[Edge](4)--(3);
                \draw[Edge](6)--(7);
                \draw[Edge](7)--(11);
                \draw[Edge](8)--(9);
                \draw[Edge](9)--(7);
                \node(r)at(5.00,3.5){};
                \draw[Edge](r)--(5);
            \end{tikzpicture}}}
        \enspace \circ_{\ColB{6}} \enspace
        \BasisE_{
            \scalebox{.70}{
            \begin{tikzpicture}[xscale=.16,yscale=.15,Centering]
                \node[Leaf](0)at(0.00,-4.50){};
                \node[Leaf](2)at(2.00,-4.50){};
                \node[Leaf](4)at(4.00,-4.50){};
                \node[Leaf](6)at(6.00,-6.75){};
                \node[Leaf](8)at(8.00,-6.75){};
                \node[Node,draw=ColB,fill=ColB!50](1)at(1.00,-2.25){};
                \node[Node,draw=ColB,fill=ColB!50](3)at(3.00,0.00){};
                \node[Node,draw=ColB,fill=ColB!50](5)at(5.00,-2.25){};
                \node[Node,draw=ColB,fill=ColB!50](7)at(7.00,-4.50){};
                \draw[Edge](0)--(1);
                \draw[Edge](1)--(3);
                \draw[Edge](2)--(1);
                \draw[Edge](4)--(5);
                \draw[Edge](5)--(3);
                \draw[Edge](6)--(7);
                \draw[Edge](7)--(5);
                \draw[Edge](8)--(7);
                \node(r)at(3.00,3){};
                \draw[Edge](r)--(3);
            \end{tikzpicture}}}
        \enspace = \enspace
        \BasisE_{
            \scalebox{.70}{
            \begin{tikzpicture}[xscale=.15,yscale=.100,Centering]
                \node[Leaf](0)at(0.00,-10.50){};
                \node[Leaf](10)at(10.00,-17.50){};
                \node[Leaf](12)at(12.00,-10.50){};
                \node[Leaf](14)at(14.00,-10.50){};
                \node[Leaf](16)at(16.00,-14.00){};
                \node[Leaf](18)at(18.00,-17.50){};
                \node[Leaf](2)at(2.00,-10.50){};
                \node[Leaf](20)at(20.00,-17.50){};
                \node[Leaf](4)at(4.00,-7.00){};
                \node[Leaf](6)at(6.00,-14.00){};
                \node[Leaf](8)at(8.00,-17.50){};
                \node[Node](1)at(1.00,-7.00){};
                \node[Node](3)at(3.00,-3.50){};
                \node[Node](5)at(5.00,0.00){};
                \node[Node](7)at(7.00,-10.50){};
                \node[Node](9)at(9.00,-14.00){};
                \node[Node,draw=ColB,fill=ColB!50](11)at(11.00,-7.00){};
                \node[Node,draw=ColB,fill=ColB!50](13)at(13.00,-3.50){};
                \node[Node,draw=ColB,fill=ColB!50](15)at(15.00,-7.00){};
                \node[Node,draw=ColB,fill=ColB!50](17)at(17.00,-10.50){};
                \node[Node](19)at(19.00,-14.00){};
                \draw[Edge](0)--(1);
                \draw[Edge](1)--(3);
                \draw[Edge](10)--(9);
                \draw[Edge](11)--(13);
                \draw[Edge](12)--(11);
                \draw[Edge](13)--(5);
                \draw[Edge](14)--(15);
                \draw[Edge](15)--(13);
                \draw[Edge](16)--(17);
                \draw[Edge](17)--(15);
                \draw[Edge](18)--(19);
                \draw[Edge](19)--(17);
                \draw[Edge](2)--(1);
                \draw[Edge](20)--(19);
                \draw[Edge](3)--(5);
                \draw[Edge](4)--(3);
                \draw[Edge](6)--(7);
                \draw[Edge](7)--(11);
                \draw[Edge](8)--(9);
                \draw[Edge](9)--(7);
                \node(r)at(5.00,4){};
                \draw[Edge](r)--(5);
        \end{tikzpicture}}}.
    \end{equation}
    %
    \item For any $s \geq 1$, the \Def{$s$-interstice operad}~\cite{Chapoton2002,CG22} is
    the nonsymmetric operad $\Interstice_s$ wherein for any $n \geq 1$, $\Interstice_s(n)$
    is the linear span of the set $[s]^n$. The set $\{\BasisE_u : u \in [s]^n, n \geq 1\}$
    is a basis of $\Interstice_s$. The partial composition of $\Interstice_s$ satisfies
    \begin{math}
        \BasisE_u \circ_i \BasisE_v
        := \BasisE_{u(1, i - 1)
        \; \Conc \; v \; \Conc \;
        u(i, \Length(u))}.
    \end{math}
    For instance, in $\Interstice_2$, we have
    \begin{math}
        \BasisE_{\ColA{121 121}} \circ_4 \BasisE_{\ColB{21}}
        =
        \BasisE_{\ColA{121} \; \ColB{21} \; \ColA{121}}.
    \end{math}
\end{itemize}
As shown in~\cite{AL2007}, $\Dup$ and $\Interstice_2$ are nonsymmetric operad quotients of
$\RightAs$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Operadic structures on packed words} \label{sec:operads_packed_words}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Right version} \label{subsec:right_version}
Let $\RightPAs$ be the graded space such that for any $n \geq 1$, $\RightPAs(n)$ is the
linear span of the set $\SetPackedWords_n$. The set $\{\BasisE_u : u \in \SetPackedWords_n,
n \geq 1\}$ is a basis of $\RightPAs$. Let us endow $\RightPAs$ with the operations
$\circ_i$ defined, for any $u \in \SetPackedWords_n[r]$, $i \in [n]$, and $v \in
\SetPackedWords_{m}[s]$, by
\begin{equation}\label{RightPAs_Def}
    \BasisE_u \circ_i \BasisE_v :=
        \BasisE_{
        \Increment{s - 1}{u(i)}(u(1, i - 1))
        \; \Conc \;
        \Increment{u(i) - 1}{0}(v)
        \; \Conc \;
        \Increment{s - 1}{u(i) - 1}(u(i + 1, n))}.
\end{equation}
For instance,
\begin{math}
    \BasisE_{\ColA{{\bf 2} 311} \ColB{\bf 2} \ColA{ {\bf 2}3}}
    \circ_5
    \BasisE_{\ColB{221}}
    =
    \BasisE_{\ColA{{\bf 2} 411} \; \ColB{332} \; \ColA{ {\bf 3}4}}.
\end{math}
Intuitively, the partial composition $\BasisE_u \circ_i \BasisE_v$ of $\RightPAs$ is similar
to the one $\RightAs$ but with the difference that the occurrences of $u(i)$ in $u$ having a
position greater than $i$ are incremented by $s - 1$ where $s$ is the maximal value of $v$.
In terms of permutation matrices, we have

\begin{scriptsize}
\begin{equation}
    \BasisE_{
    \begin{bmatrix}
        \ColA{0} & \ColA{0} & \ColA{1} &  \ColA{1} & 0 & \ColA{0} & \ColA{0} \\
        \ColA{1} & \ColA{0} & \ColA{0}  & \ColA{0} & \ColB{\bf 1} & \ColA{1} & \ColA{0} \\
        \ColA{0} & \ColA{1} & \ColA{0} &  \ColA{0} & 0 & \ColA{0} & \ColA{1}
    \end{bmatrix}}
    \circ_5
    \BasisE_{
    \begin{bmatrix}
        \ColB{0} & \ColB{0} & \ColB{1} \\
        \ColB{1} & \ColB{1} & \ColB{0}
    \end{bmatrix}}
    =
    \BasisE_{
    \begin{bmatrix}
        \ColA{0} & \ColA{0} & \ColA{1} & \ColA{1} & 0 & 0 & 0 & \ColA{0} & \ColA{0} \\
        \ColA{1} & \ColA{0} & \ColA{0} & \ColA{0} & \ColB{0} & \ColB{0} & \ColB{1} & 0 & 0 \\
        0 & 0 & 0 & 0 & \ColB{1} & \ColB{1} & \ColB{0} & \ColA{1} & \ColA{0} \\
        \ColA{0} & \ColA{1} & \ColA{0} & \ColA{0} & 0 & 0 & 0 & \ColA{0} & \ColA{1}
    \end{bmatrix}}.
\end{equation}
\end{scriptsize}

\begin{Theorem} \label{thm:rigth_packed_word_associative_operad}
    The graded space $\RightPAs$, endowed with the partial composition maps $\circ_i$, is a
    nonsymmetric unital operad.
\end{Theorem}

We call $\RightPAs$ the \Def{right associative operad of packed words}. Remark that
in~\cite{Gir2015} and~\cite{McCS2003}, other operads involving the same space of packed
words are constructed. The operads $\RightPAs$ and $\PW$, which is defined
in~\cite{Gir2015}, are not isomorphic. Indeed, a simple inspection shows that any minimal
generating set of $\PW$ contains two elements of arity $3$ while any minimal generating set
$\RightPAs$ has no element of this arity.

For any $u \in \mathfrak{P}_k$ and $i \in [k]$, the \Def{record} $\text{rec}_i(u)$ of $u$ at
$i$ is $\text{rec}_i(u):=\text{st}(u)(i)$. For example, taking $u=2311{\bf 2}23$ and $i=5$,
we have $\st(u)=3612{\bf 4}57$, so $\text{rec}_5(u)=\st(u)(5)=4$. In what follows, we
consider that the maps $\record$ and $\st$ are extended linearly on the graded space
$\RightPAs$ through its $\BasisE$-basis. In the same way, we extend linearly the action
$\cdot$ of~\eqref{equ:right_action_permutation} on $\RightPAs$ through its $\BasisE$-basis.

As shown by the next result, the operad $\RightPAs$ relates well with $\RightAs$ and
$\Interstice_2$.

\begin{Theorem} \label{thm:decopmposition_of_rigth_packed_word_associative_operad}
    Let $n \in \mathbb{N}$, $u \in \mathfrak{P}_n$, and $v \in \mathfrak{P}$. For any $i \in
    [n]$,
    \begin{equation}\label{Relating_B}
        \BasisE_u \circ_i \BasisE_v
        =
        \left(\BasisE_{\chi(u)} \circ_{\record_i(u)} \BasisE_{\chi(v)}\right)
        \cdot B_i(\st(u), \st(v)),
    \end{equation}
    where the partial composition $\circ_{\record_i(u)}$ is the one of $\Interstice_2$. In
    particular, $\chi(\BasisE_u \circ_i \BasisE_v) = \BasisE_{\chi(u)} \circ_{\record_i(u)}
    \BasisE_{\chi(v)}$ and $\st\left(\BasisE_u \circ_i \BasisE_v\right) =
    \BasisE_{B_i(\st(u),\, \st(v))}$.
\end{Theorem}

Following with our example let $u=2311{\bf 2}23$ and $v = 221$. Notice that $\st(u) =
3612457$, so that $\record_5(u)=4$. Also, $\st(v)= 231$. Therefore,
\begin{equation}
    \left(\BasisE_{\chi(u)} \circ_{\record_i(u)} \BasisE_{\chi(v)}\right)
        \cdot B_i(\st(u), \st(v))
        =
        \BasisE_{112233344} \cdot 381256479 = \BasisE_{241133234},
\end{equation}
which agrees with our previous example at the beginning of \Cref{subsec:right_version}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Left version} \label{subsec:left_version}
Let $\LeftPAs$ be the graded space such that for any $n \geq 1$, $\LeftPAs(n)$ is the linear
span of the set $\SetPackedWords[n]$. The set $\{\BasisE_u : u \in \SetPackedWords[n], n
\geq 1\}$ is a basis of $\LeftPAs$. Let us endow $\LeftPAs$ with the operations $\circ_i$
defined, for any $u \in \SetPackedWords[n]$, $i \in [n]$, and $v \in \SetPackedWords[m]$, by
\begin{equation}\label{LeftPAs_Def}
    \BasisE_u \circ_i \BasisE_v
    = \BasisE_{\Increment{m - 1}{i}(u) \; \Replace_i \; \Increment{i - 1}{0}(v)}.
\end{equation}
Intuitively, if
\begin{math}
    \BasisE_w = \BasisE_u \circ_i \BasisE_v,
\end{math}
then $w$ is the word obtained by increasing by $m - 1$ the letters of $u$ which are greater
than $i$, and by replacing all its occurrences of $i$ by the word obtained by increasing by
$i - 1$ all letters of $v$. For instance,
\begin{math}
    \BasisE_{\ColA{2} \ColB{3} \ColA{1} \ColB{3} \ColA{14}}
    \circ_{\ColB{3}}
    \BasisE_{\ColB{212}}
    =
    \BasisE_{\ColA{2} \; \ColB{434} \; \ColA{1} \; \ColB{434} \; \ColA{15}}.
\end{math}
In terms of permutation matrices, we have

\begin{scriptsize}
\begin{equation}
    \BasisE_{
    \begin{bmatrix}
        \ColA{0} & 0 & \ColA{1} & 0 & \ColA{1} & \ColA{0} \\
        \ColA{1} & 0 & \ColA{0} & 0 & \ColA{0} & \ColA{0} \\
        \ColA{0} & \ColB{\bf 1} & \ColA{0} & \ColB{\bf 1} & \ColA{0} & \ColA{0} \\
        \ColA{0} & 0 & \ColA{0} & 0 & \ColA{0} & \ColA{1}
    \end{bmatrix}}
    \circ_3
    \BasisE_{
    \begin{bmatrix}
        \ColB{0} & \ColB{1} & \ColB{0} \\
        \ColB{1} & \ColB{0} & \ColB{1}
    \end{bmatrix}}
    =
    \BasisE_{
    \begin{bmatrix}
        \ColA{0} & 0 & 0 & 0 & \ColA{1} & 0 & 0 & 0 & \ColA{1} & \ColA{0} \\
        \ColA{1} & 0 & 0 & 0 & \ColA{0} & 0 & 0 & 0 & \ColA{0} & \ColA{0} \\
        \ColA{0} & \ColB{0} & \ColB{1} & \ColB{0} & \ColA{0} & \ColB{0} & \ColB{1}
            & \ColB{0} & \ColA{0} & \ColA{0} \\
        \ColA{0} & \ColB{1} & \ColB{0} & \ColB{1} & \ColA{0} & \ColB{1} & \ColB{0}
            & \ColB{1} & \ColA{0} & \ColA{0} \\
        \ColA{0} & 0 & 0 & 0 & \ColA{0} & 0 & 0 & 0 & \ColA{0} & \ColA{1}
    \end{bmatrix}}.
\end{equation}
\end{scriptsize}%
Let us also endow $\LeftPAs$ with the right action $\SymmetricAction$ of the symmetric
groups such that, for any $u \in \SetPackedWords[n]$ and $\sigma \in \SetPermutations_n$,
\begin{equation}\label{Example_right_action}
    \BasisE_u \SymmetricAction \sigma
    = \BasisE_{\sigma^{-1}(u(1)) \; \dots \; \sigma^{-1}(u(\Length(u)))}.
\end{equation}
For instance,
\begin{math}
    \BasisE_{1411232} \SymmetricAction 3142 = \BasisE_{2322414}.
\end{math}

\begin{Theorem} \label{thm:left_packed_word_associative_symmetric_operad}
    The graded space $\LeftPAs$, endowed with the partial composition maps $\circ_i$ and the
    action~$\SymmetricAction$ of the symmetric groups, is a symmetric operad.
\end{Theorem}

We call $\LeftPAs$ the \Def{left associative operad of packed words}. Observe that this
operad is not combinatorial since for any $n \geq 1$, there are infinitely many packed words
with $n$ as maximal value. However, as we shall see in the next sections, this operad admits
interesting quotient operads which are combinatorial.
\medbreak

Given a packed word $w \in \mathfrak{P}_n[k]$, we define the \Def{set-partition type} of $w$
(also called \emph{initial permutation of $w$} in \cite{BGLPV2023}) as the packed word
$\Psi(w):=w \cdot \First{1}(w)$, where $\SymmetricAction$ is the right action and
$\First{1}(w)$ is the permutation in $\mathfrak{S}_k$ obtained by keeping from $w$ the first
appearance of each letter (see \Cref{QuotientsCombinatorial}). Our terminology is due
to the fact that $\Psi(w)$ can be related to a particular set-partition as follows. A
\Def{set-composition} of $[n]$ is a set-partition endowed with a total order on its set of
blocks. We denote by $\Sigma[n]$ and $\Pi[n]$ the sets of set-compositions and
set-partitions of $[n]$, respectively. There is a map $\text{fgt}: \Sigma[n]\to \Pi[n]$ that
forgets the order of the blocks. If $F=(B_1, B_2, \hdots, B_k) \in \Sigma[n]$, the map
$\Theta: w \mapsto F$ given by the condition $(i \in B_j \leftrightarrow w_i=j)$ defines a
bijection $\Theta: \mathfrak{P}_n[k] \to \Sigma_k[n]$. Hence, the set-composition
$\Theta(\Psi(w)) = (B_1, B_2, \hdots, B_k)$ satisfies $\min(B_1) < \cdots < \min(B_k)$. In
this sense, we can identify $\Psi(w)$ with a set-partition. For instance, if $u =231314$ and
$v=212$, then $\First{1}(u) = 2314$, $\First{1}(v) = 21$, so
\begin{math}
    \Psi(u) = 231314 \cdot 2314= 123234 \leftrightarrow (1, 24, 35,6)
\end{math}
and
\begin{math}
    \Psi(v) = 212 \cdot 21 = 121 \leftrightarrow (13, 2).
\end{math}

The following result is a ``leftist-analog'' of Hivert's Lemma \eqref{stHiv2}.

\begin{Lemma}\label{LEFTstHiv2}
     For any $w \in \mathfrak{P}_n[k]$,
     \begin{math}
         w = \Psi(w)\cdot \First{1}(w)^{-1},
     \end{math}
    with respect to the right action of permutations. Moreover, $\First{1}(w)$ is the
    smallest permutation in $\mathfrak{S}_n$ for the left weak order satisfying the above
    property.
\end{Lemma}

For any $u \in \mathfrak{P}[n]$ and $i \in [n]$, the \Def{co-record} $\text{crec}_i(u)$ of
$u$ at $i$ is $\text{crec}_i(w)$$:=$$\First{1}(u)^{-1}(i)$. For example, taking $u=231314$ and
$i=3$, we have $\First{1}(u)=2314$, so $\text{crec}_3(u)=2$. As before, we consider that the
maps $\text{crec}$ and $\First{1}$ are extended linearly on the graded space $\LeftPAs$
through its $\BasisE$-basis. An analogue result to Theorem
\eqref{thm:decopmposition_of_rigth_packed_word_associative_operad} is presented now.

\begin{Theorem} \label{thm:decopmposition_of_left_packed_word_associative_operad}
    Let $n \in \mathbb{N}$, $u \in \mathfrak{P}[n]$, and $v \in \mathfrak{P}$. For any $i
    \in [n]$,
    \begin{equation}\label{Left_Relating_B}
        \BasisE_u \circ_i \BasisE_v
        =
        \left(\BasisE_{\Psi(u)} \circ_{\text{crec}_i(u)} \BasisE_{\Psi(v)}\right)
        \cdot B_i(\First{1}(u)^{-1}, \First{1}(v)^{-1}),
    \end{equation}
    In particular, $\Psi(\BasisE_u \circ_i \BasisE_v) = \BasisE_{\Psi(u)}
    \circ_{\text{crec}_i(u)} \BasisE_{\Psi(v)}$ and $\First{1}\left(\BasisE_u \circ_i
    \BasisE_v\right) = \BasisE_{B_i(\First{1}(u)^{-1},\, \First{1}(v))^{-1}}$.
\end{Theorem}

Following with our example let $u=2 {\bf 3} 1 {\bf 3}14$ and $v = 212$. From here,
\begin{small}
\begin{equation}
    \left(\BasisE_{\Psi(u)} \circ_{\text{crec}_3(u)} \BasisE_{\Psi(v)}\right)
        \cdot B_3(\First{1}(u)^{-1}, \First{1}(v)^{-1})
        = \BasisE_{1{\bf 232}4{\bf 232}45} \cdot 24315^{-1}
        = \BasisE_{2{\bf 434}1{\bf 434}15},
\end{equation}
\end{small}
which agrees with our previous example at the beginning of \Cref{subsec:left_version}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Operadic quotients} \label{sec:quotients}
In this section, we regard the operad $\LeftPAs$ as a set-operad through its
$\BasisE$-basis. This is possible since the composition of two elements of the
$\BasisE$-basis produces a single element of the $\BasisE$-basis. For this reason, we shall
write here $u$ instead of $\BasisE_u$ for any $u \in \SetPackedWords$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Permutative congruences}
Let $P$ be a predicate on $\PositiveN^* \times \PositiveN \times \PositiveN \times
\PositiveN^*$. From $P$, we define the binary relation $\AdjacencyRelation_P$ on
$\PositiveN^*$ satisfying $uabv \AdjacencyRelation_P ubav$ for any $u, v \in \PositiveN^*$
and $a, b \in \PositiveN$ such that $P(u, a, b, v)$ holds. Let also $\Equiv_P$ be the
reflexive, symmetric, and transitive closure of $\AdjacencyRelation_P$. The predicate $P$ is
\begin{enumerate}[fullwidth]
    \item \Def{compatible with relabeling} if for any $u, v \in \PositiveN^*$ and $a, b \in
    \PositiveN$, $P(u, a, b, v)$ implies that $P\Par{f(u), f(a), f(b), f(v)}$ where $f :
    \PositiveN \to \PositiveN$ is any strictly monotone map;
    %
    \item \Def{compatible with subwords} if for any $u, v \in \PositiveN^*$ and $a, b \in
    \PositiveN$,  $P(u, a, b, v)$ implies that $P\Par{u', a, b, v'}$ for any $u', v' \in
    \PositiveN^*$ such that $u$ is a subword of $u'$ and $v$ is a subword of~$v'$.
\end{enumerate}
Observe in particular that when $P$ is compatible with subwords, $\Equiv_P$ is a monoid
congruence of the free monoid on $\PositiveN$. Here are some examples of predicates
compatible with relabeling and subwords, where $u, v \in \PositiveN^*$ and $a, b, c, d \in
\PositiveN$:
\begin{itemize}[fullwidth]
    \item Let $\PredicateFinest$ be the \Def{finest predicate} for which
    $\PredicateFinest(u, a, b, v)$ never holds.
    %
    \item Let $\PredicateComm$ be the \Def{commutative predicate}, for which
    $\PredicateComm(u, a, b, v)$ always holds.
    %
    \item Let $\PredicateSylv$ be the \Def{sylvester predicate}, satisfying
    $\PredicateSylv(u, a, c, v)$ if there is $b$ in $v$ such that $a \leq b < c$. The
    equivalence relation $\Equiv_{\PredicateSylv}$ is the \Def{sylvester
    congruence}~\cite{HNT05}, providing an alternative construction of the Loday--Ronco Hopf
    algebra.
    %
    \item Let $\PredicateHypoplactic$ be the \Def{hypoplactic predicate}, satisfying
    $\PredicateHypoplactic(u, a, c, v)$ if there is $b$ in $u$ such that $a < b \leq c$ or
    there is $b$ in $v$ such that $a \leq b < c$. The equivalence relation
    $\Equiv_{\PredicateHypoplactic}$ is the \Def{hypoplactic congruence}~\cite{KT97},
    providing a construction of the Hopf algebra $\NCSF$ of noncommutative symmetric
    functions.
    %
    \item Let $\PredicateBaxter$ be the \Def{Baxter predicate}, satisfying
    $\PredicateBaxter(u, a, c, v)$ if there is $b$ in $u$ and $b'$ in $v$ such that $a \leq
    b' < b \leq c$ or $a < b \leq b' < c$. The equivalence relation
    $\Equiv_{\PredicateBaxter}$ is the \Def{Baxter congruence}~\cite{Gir12}, providing a
    construction of the Baxter Hopf algebra which admits the Loday--Ronco Hopf algebra as
    quotient.
\end{itemize}
In contrast, the \Def{plactic predicate} $\PredicatePlactic$, satisfying
$\PredicatePlactic(u, a, c, v)$ if $v$ is nonempty and its first letter $b$ is such that $a
\leq b < c$, or $u$ is nonempty and its last letter $b$ is such that $a < b \leq c$, is not
compatible  with subwords. The equivalence relation $\Equiv_{\PredicatePlactic}$ enjoys a
lot of properties (see for instance~\cite{DHT02} for properties related to the construction
of Hopf algebras) but does not play any role in this operadic context.

\begin{Theorem} \label{thm:left_packed_word_operad_permutative_congruences}
    If $P$ is a predicate compatible with relabeling and subwords, then $\Equiv_P$ is a
    nonsymmetric operad congruence of $\LeftPAs$.
\end{Theorem}

When $P$ satisfies the prerequisites of
\Cref{thm:left_packed_word_operad_permutative_congruences}, $\Equiv_P$ is a
\Def{permutative congruence}. Let us denote by $\theta_P : \LeftPAs \to \LeftPAs$ the map
sending any $u \in \LeftPAs$ to the minimal word w.r.t.\ the lexicographic order of the
$\Equiv_P$-equivalence class of $u$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Quotients forming combinatorial operads}\label{QuotientsCombinatorial}
For any $k \geq 1$, let $\First{k} : \LeftPAs \to \LeftPAs$ be the map sending any $u \in
\LeftPAs$ to the packed word obtained by deleting any occurrence of a letter $a$ provided
that there are at least $k$ occurrences of $a$ on its left. For instance,
\begin{math}
    \First{2}(421431 \ColB{4} \ColB{1} 2) = 4214312.
\end{math}
Let us denote by $\Equiv_k$ the equivalence relation on $\LeftPAs$ satisfying, for any $u, v
\in \LeftPAs$, $u \Equiv_k v$ if $\First{k}(u) = \First{k}(v)$.

\begin{Proposition} \label{thm:left_packed_word_operad_first_congruences}
    For any $k \geq 1$, the equivalence relation $\Equiv_k$ is a nonsymmetric operad
    congruence of $\LeftPAs$.
\end{Proposition}

Since for any $k \geq 1$ and $n \geq 1$, there are finitely packed words having $n$ as
maximal value and where each letter appears at most $k$ times, the operad $\LeftPAs
/_{\Equiv_k}$ is combinatorial. The quotient $\LeftPAs /_{\Equiv_1}$ is the left associative
operad of permutations.

A predicate $P$ is \Def{compatible with $\First{k}$} if for any $u, v \in \PositiveN^*$ and
$a, b \in \PositiveN$, $P(u, a, b, v)$ implies that $P(\First{k}(u), a, b, \First{k}(v))$.
Besides, $P$ is \Def{right-trivial} if for any $u, v \in \PositiveN^*$ and $a, b \in
\PositiveN$, $P(u, a, b, v)$ implies that $P(u, a, b, \epsilon)$.

\begin{Theorem} \label{thm:commutation_projection_permutative_first_k}
    Let $k \geq 1$. If $P$ is a predicate compatible with subwords and $\First{k}$, and $P$
    is right-trivial, then the maps $\First{k}$ and $\theta_P$ commute for any $k \geq 1$.
\end{Theorem}

Let us denote by $\Equiv_{P, k}$ the equivalence relation on $\LeftPAs$ satisfying, for any
$u, v \in \LeftPAs$, $u \Equiv_{P, k} v$ if $\theta_P\Par{\First{k}(u)} = \theta_P
\Par{\First{k}(v)}$. By \Cref{thm:commutation_projection_permutative_first_k},
$\Equiv_{P, k}$ is a nonsymmetric operad congruence of $\LeftPAs$. By construction, the set
$\theta_P\Par{\First{k}(\LeftPAs)}$ is a set of representatives of the quotient $\LeftPAs
/_{\Equiv_{P, k}}$. We shall identify $\theta_P\Par{\First{k}(\LeftPAs)}$ with such set of
representatives. Considering this, the partial composition map of $\LeftPAs /_{\Equiv_{P,
k}}$ satisfies, for any $u, v \in \LeftPAs /_{\Equiv_{P, k}}$ and $i \in [|u|]$,
\begin{math}
    u \circ_i v = \theta_P\Par{\First{k}(u \circ_i v)}
\end{math}
where the second occurrence of $\circ_i$ is the partial composition map of $\LeftPAs$.

Let us review some quotients of $\LeftPAs$ obtained via such congruences.

\textbf{Finest predicate.}
The finest predicate $\PredicateFinest$ is compatible with $\First{k}$ for any $k \geq 1$
and is right-trivial. In particular, the nonsymmetric operad $\LeftPAs
/_{\Equiv_{\PredicateFinest, 1}}$ is the left associative operad $\LeftAs$. For this reason,
the operads $\LeftPAs /_{\Equiv_{\PredicateFinest, k}}$ are generalizations of the left
associative operad. For any $k \geq 1$ and any $n \geq 1$, $\LeftPAs
/_{\Equiv_{\PredicateFinest, 1}}(n)$ is the set of words such that each letter between $1$
and $n$ appears from $1$ to $k$ times. The sequences of the dimensions of $\LeftPAs
/_{\Equiv_{\PredicateFinest, 2}}$ begins with $2$, $14$, $222$, $6384$, $291720$,
$19445040$, $1781750880$ and forms Sequence~\OEIS{A105749} of~\cite{oeis}. The sequences of
the dimensions of $\LeftPAs /_{\Equiv_{\PredicateFinest, 3}}$ begins with $3$, $62$, $5052$,
$1087104$, $487424520$, $393702654960$ and forms Sequence~\OEIS{A144422} of~\cite{oeis}.

\textbf{Commutative predicate.}
The commutative predicate $\PredicateComm$ is compatible with $\First{k}$ for any $k \geq 1$
and is right-trivial. In particular, the nonsymmetric operad $\LeftPAs
/_{\Equiv_{\PredicateComm, 1}}$ is the nonsymmetric associative operad. For this reason, the
operads $\LeftPAs /_{\Equiv_{\PredicateComm, k}}$ are generalizations of the nonsymmetric
associative operad. For any $k \geq 1$ and any $n \geq 1$, $\LeftPAs
/_{\Equiv_{\PredicateComm, k}}(n)$ is the set of weakly increasing words such that each
letter between $1$ and $n$ appears from $1$ to $k$ times. Hence, the dimension of $\LeftPAs
/_{\Equiv_{\PredicateComm, k}}(n)$ is~$k^n$.

\textbf{Sylvester predicate.}
The sylvester predicate $\PredicateSylv$ is compatible with $\First{k}$ for any $k \geq 1$
but is not right-trivial. Nevertheless, by setting $\PredicateCoSylv$ as the predicate
satisfying $\PredicateCoSylv(u, a, b, v)$ if and only if $\PredicateCoSylv(v, a, b, u)$,
$\PredicateCoSylv$ is right-trivial. In particular, the nonsymmetric operad $\LeftPAs
/_{\Equiv_{\PredicateCoSylv, 1}}$ is isomorphic to the duplicial operad $\Dup$. This can be
shown by computing the nontrivial relations satisfied by the generators $12$ and $21$ of
$\LeftPAs /_{\Equiv_{\PredicateCoSylv, 1}}$ and showing that they are analog to the
nontrivial relations satisfied by the generators of $\Dup$. For this reason, the operads
$\LeftPAs /_{\Equiv_{\PredicateCoSylv, k}}$ are generalizations of the duplicial operad. By
using some results of~\cite{HNT05}, for any $k \geq 1$ and any $n \geq 1$, $\LeftPAs
/_{\Equiv_{\PredicateCoSylv, k}}(n)$ is the set of words such that each letter between $1$
and $n$ appears from $1$ to $k$ times and which avoid the patterns $231$ and $221$. The
sequence of the dimensions of $\LeftPAs /_{\Equiv_{\PredicateCoSylv, 2}}$ begins with $2$,
$10$, $66$, $498$, $4066$, $34970$, $312066$, $2862562$.

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\section{Monomial bases and Dynkin idempotents}
The weak  order on permutations has several analogues for packed words. The \Def{right weak
order} for packed words is defined via the following covering relation: for $u,v \in
\SetPackedWords_k$ $u \prec_r v$ if and only if $v = u \cdot \tau$, for some transposition
$\tau \in \SetPermutations_k$ and $\Inversions(v) = \Inversions(u)+1$. Given $u \in
\SetPackedWords$, the \Def{set of left-inversions} $\text{Inv}_\ell(u)$ of $u$ is the set of
pairs $(i,j)$ such that $i<j$ and all appearances of $i$ in $u$ occur after all appearances
of $j$ in $u$. The \Def{left weak order} of packed words is defined via the following
covering relation: for $u,v \in \SetPackedWords[n]$, $u \prec_\ell v$ if and only if $v
=\tau \cdot u$, for some transposition $\tau \in \SetPermutations_n$ and
$|\text{Inv}_\ell(v)| =  |\text{Inv}_\ell(u)|+1$. See~\cite{BGLPV2023} for relevant
bibliography related to these orders.

\textbf{Composition of monomial bases.}
For every packed word $u$, consider the new elements $\BasisF_u, \BasisM_u$ in $\RightPAs$
(resp. $\BasisG_u$, $\BasisN_u$ in $\LeftPAs$), defined implicitly as:
\begin{equation}
    \BasisE_u = \sum_{v \leq_r u}\BasisF_v
    \quad \text{and} \quad
    \BasisF_u = \sum_{u \leq_\ell v} \BasisM_v
    \quad \left(\text{resp}. \quad \BasisH_u = \sum_{v \leq_\ell u}\BasisG_v
    \quad \text{and} \quad
    \BasisG_u = \sum_{u \leq_r v} \BasisN_v \right).
\end{equation}
For $\BasisX \in \{\BasisF, \BasisM\}$ and $\BasisY \in \{\BasisG, \BasisN\}$, the sets
$\{\BasisX_u : u \in \SetPackedWords_n, n \geq 0\}$ and $\{\BasisY_u : u \in
\SetPackedWords[n], n \geq 0\}$ are linear bases for the Hopf algebras $\WQSym$ and
$\WQSym^*$ of packed words, respectively (see, for example, \cite{BGLPV2023}). The
$\BasisM$-basis and $\BasisN$-basis, called \emph{monomial bases}, are of particular
interest as they provide a simple description of the primitive spaces of $\WQSym$ and
$\WQSym^*$. Following the terminology of \cite{AM2017}, we call the former primitive space
the \Def{space of Zie elements}. In the case of  permutations, a \Def{Lie element} is an
element of the primitive space of $\FQSym$, the Hopf algebra of
permutations~\cite{MR1995,DHT02}. It is straightforward to show that the bases $\BasisF$ and
$\BasisG$ satisfy the same internal operations \eqref{RightPAs_Def} and \eqref{LeftPAs_Def}
of the $\BasisE$-basis in $\RightPAs$ and $\LeftPAs$, respectively.

For any $m,n \in \mathbb{N}$ and $i \in [m]$, let the maps
\begin{math}
    B_i^\ell: \mathfrak{P}[m] \times \mathfrak{P}[n] \to \mathfrak{P}[m+n-1]
\end{math}
and
\begin{math}
    B_i^r: \mathfrak{P}_m \times \mathfrak{P}_n \to \mathfrak{P}_{m+n-1}
\end{math}
defined as the set-theoretical operations coming from the left and right operads
\eqref{LeftPAs_Def} and \eqref{RightPAs_Def}, respectively, defined on the $\BasisE$-basis.
Here, the sets $\mathfrak{P}[m] \times \mathfrak{P}[n]$ and $\mathfrak{P}_m \times
\mathfrak{P}_n$ are endowed with the partial orders obtained by taking the Cartesian product
of the right and left weak order on packed words, respectively.

\begin{Theorem}
\begin{enumerate}[fullwidth]
    \item[$(a)$] The maps $B_i^\ell$ and $B_i^r$ are order-preserving, with respect to the
    right and left weak orders, respectively. Moreover, they possess right adjoints.
    %
    \item[$(b)$] The left (resp. right) operadic composition of the $\BasisN$-basis is
    $\BasisN$-positive (resp. for the $\BasisM$-basis).
\end{enumerate}
\end{Theorem}

For instance, $\BasisM_{112} \circ_1 \BasisM_{11} = \BasisM_{1123} + \BasisM_{1132} +
\BasisM_{2213} + \BasisM_{3312}$ (which is not an interval for $\leq_\ell$), while
$\BasisN_{112} \circ_1 \BasisN_{11} = \BasisN_{11112}+ \BasisN_{11121}$ (which is an
interval for $\leq_r$).

Part $(b)$ of the above Theorem is a direct consequence of part $(a)$. In \cite{AL2007}, the
authors show a stronger result for permutations: the right-operadic composition of the basis
$\BasisM$ is a sum over an interval for the left-weak order on permutations. This surprising
phenomenon is due to the fact that the fiber of the right adjoint for $B_i^r$ in this case
is always an interval. For our more general setting to packed words, this is no longer the
case for the $\BasisM$-basis. However, computational evidence leads to the following
conjecture.

\begin{Conjecture}
    The fiber of the right adjoint for the map $B_i^\ell$ on packed words is always an
    interval. In particular, the left-operadic composition of the $\BasisN$ basis is a sum
    over an interval of the left weak order for packed words.
\end{Conjecture}

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\textbf{An analogue of Dynkin idempotents to Zie.} \label{sec:dynkin}
Given $u , v \in \SetPackedWords$, define a new operation on $\RightPAs$ as
\begin{equation}
    \left\{\BasisF_u,  \BasisF_v\right\}
    := (\BasisM_{12} \circ_2 \BasisF_v)\circ_1 \BasisF_u.
\end{equation}
For instance,
\begin{math}
    \{\BasisF_{11}, \BasisF_{121}\}
    = ((\BasisF_{12}-\BasisF_{21})\circ_2
    \BasisF_{121})\circ_1 \BasisF_{11} = \BasisF_{11232} - \BasisF_{33121}.
\end{math}
Remark that this operation is not the induced Lie bracket of $\WQSym$; while $\{\BasisF_1,
\BasisF_1\}= \BasisF_{12} - \BasisF_{21}$, we have $[\BasisF_1, \BasisF_1]=0$. If $u \in
\SetPackedWords[m]$ and $v \in \SetPackedWords[n]$, it is straightforward to show that
$\left\{\BasisF_u,  \BasisF_v\right\} = \BasisF_{u/v} - \BasisF_{u\backslash v}$, where $u/v
:= u\Increment{m}{1}(v)$ and $u \backslash v:= \Increment{n}{1}(u)v$.

Now, given $k \geq 1$, the \Def{Dynkin idempotent} $\frac{1}{k}\, \theta_k$ is defined as
the unique element in $\RightAs(k)$ for which $\theta_k \cdot a_1a_2a_3 \cdots a_k = [\cdots
[[a_1,a_2], a_3], \cdots, a_k]$ is the $k-1$ left nested commutator bracket in the  vector
space $\mathbb{K}\langle \mathbb{N} \rangle$, for any word $a_1a_2a_3 \cdots a_k \in
\mathbb{K}\langle \mathbb{N} \rangle$. Here, we consider the natural right action of
permutations (as in \eqref{Example_right_action}). It is shown in \cite[Lemma 5.2]{AL2007}
that we can express $\theta_k$ as
\begin{math}
    \theta_k = \{ \dots \{ \{\BasisF_1, \BasisF_1\}, \BasisF_1 \}, \dots , \BasisF_1\}
\end{math}
(there are $k-1$ left nested brackets). Given a composition $\mathsf{c}=c_1c_2\cdots c_n \in
\mathbb{N}^*$ of sum $k$, we define the \Def{$c$-Dynkin's idempotent}
$\frac{1}{k}\theta^{\mathsf{c}}_k$, where
\begin{math}
    \theta_k^{\mathsf{c}}:= \{ \dots \{ \{\BasisF_{\text{id}_{c_1}},
    \BasisF_{\text{id}_{c_2}}\}, \BasisF_{\text{id}_{c_3}} \}, \dots,
    \BasisF_{\text{id}_{c_n}}\}
\end{math}
and $\text{id}_r$ denotes the packed word $11\cdots 1$ of length $r$. More generally, let
$\text{id}(\mathsf{c}) \in \SetPackedWords_k[n]$ be the non-decreasing packed word with
$c_r$ copies of $r$, for every $r \in [n]$.
\begin{Theorem}
    For any $k \geq 1$, $\theta_k^{\mathsf{c}}$ is a Zie element in $\WQSym$, with
    $\BasisM$-expansion given by
    \begin{equation}
        \theta_k^{\mathsf{c}}
        = \sum_{\substack{
            \text{id}(\mathsf{c}) \; \leq_\ell \; u \\
            u = \text{id}\left(c_1\right) 2 \; \circ_2 \; u'}}
        \BasisM_u.
    \end{equation}
\end{Theorem}

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\end{document}

