%% if you are submitting an initial manuscript then you should have submission as an option here
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\documentclass[finalversion]{FPSAC2025}
\articlenumber{144}
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\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}

\usepackage{lipsum}

\usepackage{hyperref}
%\hypersetup{
	%colorlinks=true,
	%linkcolor=blue,
	%citecolor=blue}


\usepackage{graphicx}
\usepackage{amsfonts,tikz,mathrsfs}
\usepackage{amsmath}
%\usepackage{caption}

\usepackage{tikz}
\usepackage{dsfont}
\usepackage{marginnote}
\usetikzlibrary{matrix, fit}
\usepackage{amssymb, shuffle}
%\usepackage{dirtytalk}
%\usepackage[rightcaption]{sidecap}
%\usepackage{youngtab}
%\usepackage{caption}
%\usepackage{relsize}
\usepackage{amssymb}
\usepackage{dirtytalk}
\usepackage{paracol}
\usepackage{multicol}
\usepackage{wrapfig}
\usepackage{tcolorbox}











\newcommand{\blambda}{\boldsymbol{\lambda}}
\newcommand{\bmu}{\boldsymbol{\mu}}
\newcommand{\bnu}{\boldsymbol{\nu}}
\newcommand{\GL}{\rm GL}
\newcommand{\UL}{\rm UL}
%\newcommand{\U}{\rm U}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\UT}{\rm UT}
\newcommand{\DT}{\rm DT}
\newcommand{\NT}{\rm NT}
\newcommand{\bbf}{\mathbb{F}}
\newcommand{\SCNPtt}{\mathbf{SC}_{\rm NPtt}}
\newcommand{\CQSym}{\mathrm{CQSym}}
\newcommand{\NSym}{\mathrm{NSym}}
\newcommand{\NCQSym}{\mathrm{NCQSym}}
\newcommand{\NCSym}{\mathrm{NCSym}}
\newcommand{\PPLP}{\mathrm{PPLP}}
\newcommand{\EPPLP}{\mathrm{EPPLP}}
\newcommand{\FQSym}{\mathrm{FQSym}}
\newcommand{\QSym}{\mathrm{QSym}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\Peak}{\mathrm{Peak}}
\newcommand{\SetComp}{\mathrm{SetComp}}
\newcommand{\OddSetComp}{\mathrm{SetCompOdd}}
\newcommand{\peak}{\mathrm{Peak}}
\newcommand{\odd}{\mathrm{Odd}}
\newcommand{\oddd}{\mathrm{ODiff}}
\newcommand{\std}{\mathrm{std}}
\newcommand{\estd}{\mathrm{estd}}
\newcommand{\Cat}{\mathrm{Cat}}
\newcommand{\NCPeak}{\mathrm{NC}\Pi}
\newcommand{\NCGamma}{\mathrm{NC}\Gamma}
\newcommand{\Des}{\mathrm{Des}}
\newcommand{\pa}{\text{-par}}
\newcommand{\comp}{\text{-comp}}
\newcommand{\ten}{10}
\newcommand{\eleven}{11}
\newcommand{\twelve}{12}
\newcommand{\bM}{\mathbf{M}}
\newcommand{\bF}{\mathbf{F}}
\newcommand{\rmrlx}{{\rm rlx}}
\newcommand{\rmsym}{{\rm sym}}
\newcommand{\bs}{{\bf s}}
\newcommand{\bx}{{\bf x}}
\newcommand{\by}{{\bf y}}
\newcommand{\arx}{\blacktriangleleft}
\newcommand{\ary}{\trianglelefteq}
\newcommand{\arz}{\lhd}
\newcommand{\bbQ}{{\mathbb{Q}}}
\newcommand{\bfF}{{\mathbf{F}}}
\newcommand{\calA}{{\mathcal{A}}}
\newcommand{\scrX}{{\mathscr{X}}}
\newcommand{\scrY}{{\mathscr{Y}}}
\newcommand{\scrE}{{\mathscr{E}}}
\newcommand{\scrF}{{\mathscr{F}}}
\newcommand{\bfone}{{\mathbf{1}}}
\newcommand{\bftwo}{{\mathbf{2}}}
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\newcommand{\bffour}{{\mathbf{4}}}
\newcommand{\bfn}{{\mathbf{n}}}
\newcommand{\bfdet}{{\mathbf{det}}}
\newcommand{\frakS}{{\mathfrak{S}}}
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\newcommand{\calB}{{\mathcal{B}}}
\newcommand{\boldeta}{{\boldsymbol{\eta}}}
\newcommand{\sort}{{\mathrm{sort}}}
\newcommand{\rmdet}{{\mathrm{det}}}














%% define your title in the usual way
\title[Peak algebra in noncommuting variables]{Peak algebra in noncommuting variables}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[F. Aliniaeifard and S. Li]{\texorpdfstring{Farid Aliniaeifard\thanks{\href{aliniaeifard.farid@gmail.com}{aliniaeifard.farid@gmail.com}}\addressmark{1,2} \and Shu Xiao Li\thanks{\href{mailto:lishuxiao@sdu.edu.cn}{lishuxiao@sdu.edu.cn}.}\addressmark{1,2}}{Farid Aliniaeifard and Shu Xiao Li}}
%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao \\ \addressmark{2}Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Qingdao}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{The well-known descent-to-peak map $\theta$ for the Hopf algebra of quasisymmetric functions, QSym, and the peak algebra $\Pi$ were originally defined by Stembridge in 1997. We define the labelled descent-to-peak map $\Theta$ and extend the notion of the peak algebra to noncommuting variables.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{quasisymmetric functions, peak algebra, descent-to-peak map}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{sample.bib}
\appto\bibfont{\setlength{\emergencystretch}{1em}}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above


\section{Introduction}

The first comprehensive study on the combinatorics of peaks was conducted by Stembridge \cite{S97}, who developed and introduced enriched $(P,\gamma)$-partitions, which is an analog to Stanley’s theory of $(P,\gamma)$-partitions, with the key distinction that the notion of peaks replaces the notion of descents in the context of linear extensions of posets. Generating functions of $(P,\gamma)$-partitions, $\Gamma(P,\gamma)$,  give the Hopf algebra of quasisymmetric functions $\QSym$, and the generating functions of enriched $(P,\gamma)$-Partitions, $\Delta(P,\gamma)$, give the peak algebra $\Pi$. Stembridge also defined the descent-to-peak algebra morphism $\theta$ from $\QSym$ to $\Pi$ where  $\Gamma(P,\gamma)$ maps to $\Delta(P,\gamma)$, and showed that the dimension of the homogenous functions of degree $n$ of $\Pi$, $\Pi_n$, is equal to the number of odd compositions, compositions whose all parts are odd, which is equal to Fibonacci number $f_n$. Moreover, he showed that restricting the map $\theta$ to symmetric functions gives the Hopf algebra of Schur's $Q$ functions. The Hopf algebra of Schur's $Q$ functions, whose bases are indexed by odd partitions, are introduced in \cite{S11} to study the projective representations of symmetric and alternating groups. Combinatorially, the Schur’s $Q$ functions are equipped with a theory of shifted tableaux, including RSK correspondence, Littlewood--Richardson rule, and jeu de taquin \cite{S87, S89, W84}. In \cite{BMSvW}, Bergeron et al. showed that the peak algebra is a Hopf algebra and also the map $\theta$ is a Hopf algebra morphism. Also, the main result of \cite{S11} by Schocker is that the peak algebra is a left co-ideal of $\QSym$ under internal comultiplication, a generalized dual Kronecker product.  

 
  It is also shown that the peak algebra corresponds to the representations of the $0$-Hecke--Clifford algebra \cite{B04}. 
Further studies revealed connections between peaks and a variety of seemingly unrelated topics, such as the generalized Dehn--Sommerville equations \cite{ABS, BMSvW0, BHvW} and the Schubert calculus of isotropic flag manifolds \cite{BMSvW, BH}. Notably, in \cite{Nym01, Nym}, the peak algebra is generalized to the Poirier--Reutenauer Hopf algebra of standard Young tableaux, which is introduced in \cite{P95}. Other generalizations can be found in \cite{A04, B06, H06}. 

The $(P,\gamma)$-partitions are the generalized chromatic functions of certain digraphs \cite{ALvW22}, and here we introduce enriched generalized chromatic functions. The enriched  $(P,\gamma)$-partitions are enriched generalized chromatic functions of certain digraphs. Now, extending generalized chromatic functions and enriched generalized chromatic functions to noncommuting variables, we define the labelled peak-to-decent map $\Theta$ from the Hopf algebra of quasisymmetric functions in noncommuting variables $\NCQSym$ to the peak algebra in noncommuting variables $\NCPeak$. This map is indeed a Hopf algebra morphism. Applying the map $\rho$, where it commutes the variables, we obtain theta map $\theta$.  We compute the values of the map $\Theta$ at the fundamental and monomial basis of $\NCQSym$. 
Then, we define the peak algebra in noncommuting variables  $\NCPeak$ and introduce Schur's $Q$ functions in noncommuting variables. 
The dimension of the homogenous functions of degree $n$ of $\NCPeak$, $\NCPeak_n$, is equal to the odd set compositions, set compositions whose all parts are of odd size, which is equal to $a_n$ where $a_n$ is the sequence A006154 in the OEIS.
 Moreover, we show that the restriction of the map $\Theta$ to symmetric functions in noncommuting variables is the Hopf algebra of Schur's $Q$ functions in noncommuting variables $\NCSym \cap \NCPeak$. They are indexed by odd set partitions.  
We present that the peak algebra in noncommuting variables is a left co-ideal of $\NCQSym$ under internal comultiplication, extending    Schocker's result in \cite{S11} that the peak algebra is a left co-ideal of $\QSym$ under internal comultiplication, a generalized dual Kronecker product.  




\section{Generalized chromatic functions, descent-to-peak map, and peak algebra}\label{sec:painting}

In this section, we present a summary of the earlier results in the language of generalized chromatic functions \cite{ALvW22}. 

\subsection{Edge-coloured digraphs and some operators}
Stanley \cite{Sta71} defined $P$-partitions by generalizing MacMahon's work on plane partitions \cite{Mac11}\footnote{For a complete history of $P$-partitions see I. M. Gessel's paper \cite{Ges16}.}. $P$-partitions can be identified as certain vertex-colourings of some family of edge-coloured digraphs. We describe this family of edge-coloured digraphs and some useful operators between them. 




\begin{tcolorbox}[colback=black!0!white, colframe=blue!0!black, boxrule=0.0mm, left=0.1pt, right=0.1pt,  width=1\textwidth]
\begin{wrapfigure}{r}{0.25\textwidth}
\begin{center} 
 \begin{tikzpicture}[scale=0.7]
 \node(a) at (0,0){$4$};
 \node(b) at (-1.5,1.5){$2$};
 \node(c) at (1.5,1.5){$1$};
 \node(d) at (-1.5,3){$3$};
 
 
 \draw[thick,->] (a)--(b);
 \draw[thick,->] (a)--(c);
 \draw[thick,->] (b)--(d);
 \end{tikzpicture} 
 \end{center}
 \end{wrapfigure}
 The Hasse diagram of a poset $P=(X,\leq)$ can be seen as a digraph whose vertices are the elements of the ground set $X$ of the poset, and there is a directed edge from $a$ to $b$ if $a\leq b$ and if there is $c\in X$ such that $a\leq c\leq b$, then either $c=a$ or $c=b$. 
% For example, the following digraph corresponds to a poset with the ground set $\{1,2,3,4\}$.
  {\bf \emph{ Throughout this paper, all digraphs are Hasse diagrams of some posets.}} 
 \end{tcolorbox} 
 
 

 An \emph{edge-coloured digraph} is a digraph whose edges are of the form $\rightarrow$ or $\Rightarrow$.
\begin{figure} \centering
  \begin{tikzpicture}[scale=0.6,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]

 \node[squarednode](a) at (0,0){};
 \node[squarednode](b) at (-1.5,1.5){};
 \node[squarednode](c) at (1.5,1.5){};
 \node[squarednode](d) at (-1.5,3){};
 
 \draw[thick,->] (a)--(b);
 \draw[thick,double,->] (a)--(c);
 \draw[thick,double,->] (b)--(d);
 \end{tikzpicture} 
   \caption{An edge-coloured digraph}\label{Fig:edge-coloured}
 \end{figure} 
 
 We now define some useful edge-coloured digraphs. Let \index{ $Q_n$ (resp. $P_n$)}
 $Q_n$ (resp. $P_n$) be the edge-coloured directed path with $n$ vertices whose all edges are of the form  $\Rightarrow$ (resp. $\rightarrow$).  
 % Let $C_n$ be the edge-coloured digraph cycle with $n$ vertices with all edges of the from $\Rightarrow$.
  $$
  \begin{tikzpicture}[
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=7mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
 \node[squarednode](1) at (1.3,0){};
  \node[squarednode](2) at (2.6,0){};
    \node[squarednode](3) at (3.9,0){};
    \node at (2.6,-0.7){$Q_{3}$};
    \draw[thick,double,->](1)--(2);
       \draw[ thick,double,->](2)--(3);
      %  \draw[ thick,double,->](3)--(3.9,0.7)--(1.3,0.7)--(1);
        % \draw[ thick,double,->](1.5,0.7)--(1);
        %  \draw[ultra thick](2.1,0.9)--(2,1.1);
 \end{tikzpicture}\quad \quad  \quad   \begin{tikzpicture}[
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=7mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
 \node[squarednode](1) at (1.3,0){};
  \node[squarednode](2) at (2.6,0){};
    \node[squarednode](3) at (3.9,0){};
    \node at (2.6,-0.7){$P_{3}$};
    \draw[thick,->](1)--(2);
       \draw[ thick,->](2)--(3);
      %  \draw[ thick,double,->](3)--(3.9,0.7)--(1.3,0.7)--(1);
        % \draw[ thick,double,->](1.5,0.7)--(1);
        %  \draw[ultra thick](2.1,0.9)--(2,1.1);
 \end{tikzpicture} 
 %\quad \quad  \quad   \begin{tikzpicture}[
%roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=7mm},
%squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum %size=5mm},]
% \node[squarednode](1) at (1.3,0){};
%  \node[squarednode](2) at (2.6,0){};
%    \node[squarednode](3) at (3.9,0){};
%    \node at (2.6,-0.7){$C_{3}$};
%    \draw[thick,double,->](1)--(2);
%       \draw[ thick,double,->](2)--(3);
%       \draw[ thick,double,->](3)--(3.9,0.7)--(1.3,0.7)--(1);
%        % \draw[ thick,double,->](1.5,0.7)--(1);
%        %  \draw[ultra thick](2.1,0.9)--(2,1.1);
 %\end{tikzpicture} 
 $$
 
 The \emph{disjoint union} of edge-coloured digraphs $G_1$ and $G_2$ with $V(G_1)\cap V(G_2)=\emptyset$, denoted $G_1\sqcup G_2$, \index{$G_1\sqcup G_2$} is an edge-coloured digraph such that 
 \begin{enumerate} 
 \item The vertex set of $G_1\sqcup G_2$ is  the disjoint union of the vertex sets of $G_1$ and $G_2$. 
 \item The edge set of $G_1\sqcup G_2$ is  the disjoint union of the edge sets of $G_1$ and $G_2$. 
 \item $a\Rightarrow b$ in $G_1\sqcup G_2$ if either $a\Rightarrow b$ in $G_1$ or in $G_2$. 
  \item $a\rightarrow b$ in $G_1\sqcup G_2$ if either $a\rightarrow b$ in $G_1$ or in $G_2$.
 \end{enumerate} 
   $$
  \begin{tikzpicture}[
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=7mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
 \node[squarednode](1) at (1.3,0){};
  \node[squarednode](2) at (2.6,0){};
    \node[squarednode](3) at (3.9,0){};
    \draw[thick,double,->](1)--(2);
       \draw[ thick,double,->](2)--(3);
 \node[squarednode](11) at (4.8,0){};
  \node[squarednode](21) at (6.1,0){};
    \node[squarednode](31) at (7.4,0){};
    \draw[thick,->](11)--(21);
       \draw[ thick,->](21)--(31);
       \node at (4.35,-0.7){$Q_3 \sqcup P_3$};
 \end{tikzpicture} 
 $$



 The \emph{solid sum} of edge-coloured directed paths $G_1$ and $G_2$, denoted by $G_1\rightarrow G_2$, \index{$G_1\rightarrow G_2$} is an edge-coloured digraph obtained by connecting the last vertex of $G_1$ to the first vertex of $G_2$ by a solid edge $\rightarrow$. 
    $$
  \begin{tikzpicture}[
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=7mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
 \node[squarednode](1) at (1.3,0){};
  \node[squarednode](2) at (2.6,0){};
    \node[squarednode](3) at (3.9,0){};
    \node at (4.6,-0.7){$Q_{3}\rightarrow  Q_{3}$};
    \draw[thick,double,->](1)--(2);
       \draw[ thick,double,->](2)--(3);
      %  \draw[ thick,double,->](3)--(3.9,0.7)--(1.3,0.7)--(1);
        % \draw[ thick,double,->](1.5,0.7)--(1);
        %  \draw[ultra thick](2.1,0.9)--(2,1.1);
 \node[squarednode](a) at (5.2,0){${}$};
  \node[squarednode](b) at (6.5,0){};
    \node[squarednode](c) at (7.8,0){};
  %  \node at (1.4,-0.7){$\sqcup P_{\{2,6,9\}}$};
    \draw[thick,double,->](a)--(b);
       \draw[ thick,double,->](b)--(c);
          \draw[ thick,double,->](b)--(c);
            \draw[ thick,->](3)--(a);
      %  \draw[ thick,double,->](3)--(3.9,0.7)--(1.3,0.7)--(1);
        % \draw[ thick,double,->](1.5,0.7)--(1);
        %  \draw[ultra thick](2.1,0.9)--(2,1.1);
 \end{tikzpicture} 
 $$
 

 \subsection{Proper colourings and generalized chromatic functions}\label{subsec:gcf}
 
 As we mentioned earlier, $P$-partitions can be identified as certain types of vertex-\-colourings of edge-coloured digraphs. We describe these types of vertex-colourings of edge-coloured digraphs, and then we construct their generating functions, which are called generalized chromatic functions. 
 

\begin{tcolorbox}[colback=black!0!white, colframe=blue!0!black, boxrule=0.0mm, left=0.1pt, right=0.1pt,  width=1\textwidth]
\begin{wrapfigure}{r}{0.25\textwidth}
%\begin{tcolorbox}[colback=black!0!white, colframe=blue!0!black, boxrule=0.01mm, left=-0.1pt, right=-0.1pt,  width=0.32\textwidth]
\centering
\begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
                 \node[squarednode,label=above:{$2$}](d) at (0,5) {};
				\node[squarednode,label=above:{$2$}](b)  at (-2,2.5) {};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {};
				\node[squarednode, label=above:{$2$}](c)  at (2,2.5) {};
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
			\end{tikzpicture}
		% \captionof{figure}{A proper colouring}\label{Fig:proper-colouring-edge-coloured}
		% \end{tcolorbox}
\end{wrapfigure}
A \emph{proper} colouring of an edge-coloured digraph $G$ is a function $$\kappa:V(G)\rightarrow \mathbb{N}=\{1,2,3,\dots\}$$ \index{$\kappa$: proper colouring of edge-coloured digraphs} such that 
\begin{enumerate}
\item If $a\Rightarrow b$, then $\kappa(a)\leq \kappa(b)$.
\item If $a\rightarrow b$, then $\kappa(a)< \kappa(b)$.
\end{enumerate} 
\end{tcolorbox}







Recall that $\bbQ[[x_1,x_2,\dots]]$ is the algebra of formal power series in infinitely many commuting variables $x=\{x_1,x_2,\dots\}$ over $\bbQ$.

The \emph{generalized chromatic function} \index{$\scrX_G$} of an edge-coloured digraph $G$ is 
$$\scrX_G=\sum_{\kappa} \prod_{v\in V(G)} x_{\kappa(v)}$$  where the sum is over all proper colourings $\kappa$ of $G$. For example, if $G$ is the edge-coloured digraph in \Cref{Fig:edge-coloured}, then 
\begin{align*}
		\begin{array}{ccccccccc}
			& \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$2$}](d) at (0,5) {};
				\node[squarednode,label=above:{$2$}](b)  at (-2,2.5) {};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {};
				\node[squarednode, label=above:{$2$}](c)  at (2,2.5) {};
			\end{tikzpicture} &
			&\begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$3$}](d) at (0,5) {};
				\node[squarednode,label=above:{$2$}](b)  at (-2,2.5) {};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {};
				\node[squarednode, label=above:{$2$}](c)  at (2,2.5) {};
			\end{tikzpicture} &
			&  \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$2$}](d) at (0,5) {};
				\node[squarednode,label=above:{$2$}](b)  at (-2,2.5) {};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {};
				\node[squarednode, label=above:{$3$}](c)  at (2,2.5) {};
			\end{tikzpicture} & 
			& \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$3$}](d) at (0,5) {};
				\node[squarednode,label=above:{$2$}](b)  at (-2,2.5) {};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {};
				\node[squarednode, label=above:{$3$}](c)  at (2,2.5) {};
			\end{tikzpicture}\\
			\scrX_G=& x_1x_2^3 & + & x_1x_2^2x_3 & + & x_1x_2^2x_3 & + & x_1x_2x_3^2 &+\cdots.
		\end{array}
	\end{align*}
	
	Generalized chromatic functions can be seen as generating functions of $P$-partitions, and so they are quasisymmetric functions.
	
	
%	\begin{Remark}
%	Generalized chromatic functions were first appeared in \cite{ALvW22}. 
%	\end{Remark}
	


	
	\subsection{Enriched colourings and enriched chromatic functions}\label{subsec:ecf}
	Stembridge \cite{S97} defined enriched $P$-partitions and used them to associate tableaux with Schur's $Q$-functions \cite{S11}\footnote{For an English reference, see I. G. Macdonald's book \cite[Chapter III, Section 8]{M95}, where he described Schur's $Q$-functions in more detail. }. 
	Enriched $P$-partitions can be identified as certain types of vertex-\-colourings of edge-\-coloured digraphs. We describe these types of vertex-{\parfillskip=0pt\par} \noindent colourings of edge-coloured digraphs, and then we construct their generating functions, which are called enriched chromatic functions. 
	
	Given an edge-coloured digraph $G$, an \emph{enriched} colouring of $G$ is a function 
	{
  \setlength{\abovedisplayskip}{.5em}
  \setlength{\belowdisplayskip}{.5em}
	$$\kappa: V(G)\rightarrow \{\dots \prec -1\prec1\prec -2\prec 2 \prec \cdots \}$$ }
	such that 
	\begin{enumerate}[itemsep=0pt,topsep=.5em]
	\item If $a\Rightarrow b$, then either $\kappa(a)\prec \kappa(b)$ or $\kappa(a)=\kappa(b)>0$. 
	\item If $a\rightarrow b$, then either $\kappa(a)\prec \kappa(b)$ or $\kappa(a)=\kappa(b)<0$.
	\end{enumerate} 
	


The \emph{enriched chromatic function} \index{$\scrE_G$} of an edge-coloured digraph $G$ is 
$$\scrE_G=\sum_{\kappa} \prod_{v\in V(G)} x_{|\kappa(v)|}$$  where the sum is over all enriched colourings $\kappa$ of $G$. For example, if $G$ is the edge-coloured digraph in \Cref{Fig:edge-coloured}, then 
\begin{align*}
		\begin{array}{ccccccccc}
			& \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$1$}](d) at (0,5) {};
				\node[squarednode,label=above:{$-1$}](b)  at (-2,2.5) {};
				\node[squarednode,label=above:{$-1$}](a)  at (0,0) {};
				\node[squarednode, label=above:{$1$}](c)  at (2,2.5) {};
			\end{tikzpicture} &
			&\begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$1$}](d) at (0,5) {};
				\node[squarednode,label=above:{$1$}](b)  at (-2,2.5) {};
				\node[squarednode,label=above:{$-1$}](a)  at (0,0) {};
				\node[squarednode, label=above:{$1$}](c)  at (2,2.5) {};
			\end{tikzpicture} &
			&\begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$2$}](d) at (0,5) {};
				\node[squarednode,label=above:{$-2$}](b)  at (-2,2.5) {};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {};
				\node[squarednode, label=above:{$1$}](c)  at (2,2.5) {};
			\end{tikzpicture} & 
			& \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$2$}](d) at (0,5) {};
				\node[squarednode,label=above:{$-2$}](b)  at (-2,2.5) {};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {};
				\node[squarednode, label=above:{$-2$}](c)  at (2,2.5) {};
			\end{tikzpicture}\\
			\scrE_G=& x_1^4 & + & x_1^4 & + & x_1^2x_2^2 & + & x_1x_2^3&+\cdots.
		\end{array}
	\end{align*}
	



\subsection{Peak algebra and descent-to-peak map}\label{subsec:peak-theta-results}

Stembridge defined the peak algebra $\Pi$
%\footnote{See \Cref{lem:ppar-to-gcf} for the justification that this space is called the peak algebra.}
 in \cite{S97} as a space spanned by the generating functions of enriched $P$-partitions. Since generating functions of enriched $P$-partitions are the enriched chromatic functions of edge-coloured digraphs and vice versa, we have that the peak algebra is spanned by the set 
$$\{\scrE_G: \text{$G$ is an edge-coloured digraph}\}.$$ 

Stembridge also defined the \emph{descent-to-peak map}
%\footnote{See \Cref{subsec:(enr)Ppar} for the justification that this map is called descent-to-peak map.} 
$\Theta_{\QSym}$; we can write it as follows, 
 \begin{align}\label{Def:ThetaMap}
\begin{array}{cccc}
\Theta_\QSym:& \QSym & \rightarrow & \Pi\\
& \scrX_G & \mapsto & \scrE_G.
\end{array} 
\end{align} 
He showed the descent-to-peak map is a surjective algebra morphism \cite[Theorem 3.1]{S97}.


\section{Generalized chromatic functions and Peak algebra in noncommuting variables, and labelled descent-to-peak map}

\subsection{Generalized chromatic functions in noncommuting variables}

A \emph{labelled edge-coloured digraph} is an edge-coloured digraph where its vertex set is a subset of $\mathbb{N}$. We usually denote the vertices of a labelled edge-coloured digraph by bold positive integers. 
\begin{figure}
\centering
  \begin{tikzpicture}[scale=0.7,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]

 \node[squarednode](a) at (0,0){$\bf 1$};
 \node[squarednode](b) at (-1.5,1.5){$\bf 2$};
 \node[squarednode](c) at (1.5,1.5){$\bf 3$};
 \node[squarednode](d) at (-1.5,3){$\bf 4$};
 
 \draw[thick,->] (a)--(b);
 \draw[thick,double,->] (a)--(c);
 \draw[thick,double,->] (b)--(d);
 \end{tikzpicture} 
   \caption{A labelled edge-coloured digraph}\label{Fig:labelled-edge-coloured}
\end{figure}
We usually use $\bfG$ \index{$\bfG$} to denote a labelled edge-coloured digraph whose underlying edge-coloured digraph is $G$. 
%For example, if we denote the labelled edge-coloured digraph in \Cref{Fig:labelled-edge-coloured} by $\bfG$, then $G$ is the edge-coloured digraph in \Cref{Fig:edge-coloured}. 



 We now define some useful labelled edge-coloured digraphs. For any set $$S=\{i_1<i_2<\dots<i_k\}$$ of positive integers,  \index{any set $S=\{i_1<i_2<\dots<i_k\}$ of positive integers}let 
 $Q_S$ (resp. $P_S$) \index{$Q_S$ (resp. $P_S$), and $S$ index set} be the labelled edge-coloured directed path with vertex set $S$ such that its coloured edges are  ${i_j}\Rightarrow {i_{j+1}}$ (resp. ${i_j}\rightarrow {i_{j+1}}$) for $1\leq j \leq k$.   
  $$
  \begin{tikzpicture}[
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=7mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
 \node[squarednode](1) at (1.3,0){${\bf 3}$};
  \node[squarednode](2) at (2.6,0){{\bf 5}};
    \node[squarednode](3) at (3.9,0){{\bf 8}};
    \node at (2.6,-0.7){$Q_{\{3,5,8\}}$};
    \draw[thick,double,->](1)--(2);
       \draw[ thick,double,->](2)--(3);
      %  \draw[ thick,double,->](3)--(3.9,0.7)--(1.3,0.7)--(1);
        % \draw[ thick,double,->](1.5,0.7)--(1);
        %  \draw[ultra thick](2.1,0.9)--(2,1.1);
 \end{tikzpicture}\quad \quad  \quad \quad \quad   \begin{tikzpicture}[
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=7mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
 \node[squarednode](1) at (1.3,0){${\bf 3}$};
  \node[squarednode](2) at (2.6,0){{\bf 5}};
    \node[squarednode](3) at (3.9,0){{\bf 8}};
    \node at (2.6,-0.7){$P_{\{3,5,8\}}$};
    \draw[thick,->](1)--(2);
       \draw[ thick,->](2)--(3);
 \end{tikzpicture} 
 $$
 \noindent
 The \emph{disjoint union} of labelled edge-coloured digraphs $\bfG_1$ and $\bfG_2$ with $V(\bfG_1)\cap V(\bfG_2)=\emptyset$, is denoted by $\bfG_1\sqcup \bfG_2$. \index{$\bfG_1\sqcup \bfG_2,\bfG_1\rightarrow \bfG_2$}
     $$
  \begin{tikzpicture}[
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=7mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
 \node[squarednode](1) at (1.3,0){$\bf 3$};
  \node[squarednode](2) at (2.6,0){$\bf 5$};
    \node[squarednode](3) at (3.9,0){$\bf 8$};
    \draw[thick,double,->](1)--(2);
       \draw[ thick,double,->](2)--(3);
 \node[squarednode](11) at (4.8,0){$\bf 2$};
  \node[squarednode](21) at (6.1,0){$\bf 6$};
    \node[squarednode](31) at (7.4,0){$\bf 9$};
    \draw[thick,->](11)--(21);
       \draw[ thick,->](21)--(31);
       \node at (4.35,-0.7){$Q_{\{3,5,8\}} \sqcup P_{\{2,6,9\}}$};
 \end{tikzpicture} 
 $$



 The \emph{solid sum} of labelled edge-coloured directed paths $\bfG_1$ and $\bfG_2$, denoted $\bfG_1\rightarrow \bfG_2$, is an edge-coloured digraph obtained by connecting the last vertex of  $\bfG_1$ to the first vertex of $\bfG_2$ by a solid edge $\rightarrow$. 
    $$
  \begin{tikzpicture}[
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=7mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
 \node[squarednode](1) at (1.3,0){${\bf 3}$};
  \node[squarednode](2) at (2.6,0){{\bf 5}};
    \node[squarednode](3) at (3.9,0){{\bf 8}};
    \node at (4.6,-0.7){$Q_{\{3,5,8\}}\rightarrow  Q_{\{2,6,9\}}$};
    \draw[thick,double,->](1)--(2);
       \draw[ thick,double,->](2)--(3);
 \node[squarednode](a) at (5.2,0){${\bf 2}$};
  \node[squarednode](b) at (6.5,0){{\bf 6}};
    \node[squarednode](c) at (7.8,0){{\bf 9}};
    \draw[thick,double,->](a)--(b);
       \draw[ thick,double,->](b)--(c);
          \draw[ thick,double,->](b)--(c);
            \draw[ thick,->](3)--(a);
 \end{tikzpicture} 
 $$
 

Recall that $\bbQ\langle \langle \bx_1,\bx_2,\dots\rangle\rangle$ is the algebra of formal power series in infinitely many noncommuting variables $\bx=\bx_1,\bx_2,\dots$ over $\bbQ$.


The \emph{generalized chromatic function in noncommuting variables} of a labelled edge-coloured digraph $\bfG$ with vertex set $[n]$ is 
$$\scrY_\bfG=\sum_{\kappa} \bx_{\kappa(1)} \bx_{\kappa(2)} \dots \bx_{\kappa(n)}$$ \index{$\scrY_\bfG$}where the sum is over all proper colourings $\kappa$ of $G$. For example, if $\bfG$ is the labelled edge-coloured digraph in \Cref{Fig:labelled-edge-coloured}, then 
\begin{align*}
		\begin{array}{ccccccccc}
			& \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$2$}](d) at (0,5) {$\bffour$};
				\node[squarednode,label=above:{$2$}](b)  at (-2,2.5) {$\bftwo$};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {$\bfone$};
				\node[squarednode, label=above:{$2$}](c)  at (2,2.5) {$\bfthree$};
			\end{tikzpicture} &
			&\begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$3$}](d) at (0,5) {$\bffour$};
				\node[squarednode,label=above:{$2$}](b)  at (-2,2.5) {$\bftwo$};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {$\bfone$};
				\node[squarednode, label=above:{$2$}](c)  at (2,2.5) {$\bfthree$};
			\end{tikzpicture} &
			&  \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$2$}](d) at (0,5) {$\bffour$};
				\node[squarednode,label=above:{$2$}](b)  at (-2,2.5) {$\bftwo$};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {$\bfone$};
				\node[squarednode, label=above:{$3$}](c)  at (2,2.5) {$\bfthree$};
			\end{tikzpicture} & 
			& \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$3$}](d) at (0,5) {$\bffour$};
				\node[squarednode,label=above:{$2$}](b)  at (-2,2.5) {$\bftwo$};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {$\bfone$};
				\node[squarednode, label=above:{$3$}](c)  at (2,2.5) {$\bfthree$};
			\end{tikzpicture}\\
			\scrY_G=& \bx_1\bx_2\bx_2\bx_2 & + & \bx_1\bx_2\bx_2\bx_3 & + & \bx_1\bx_2\bx_3\bx_2 & + & \bx_1\bx_2\bx_3\bx_3 &+\cdots.
		\end{array}
	\end{align*}
	


\subsection{Enriched chromatic functions in noncommuting variables}\label{subsec:enriched-chromatic-functions} 
The \emph{enriched chromatic function} in noncommuting variables of a labelled edge-coloured digraph $\bfG$ with vertex set $[n]$ is 
$$\scrF_\bfG=\sum_{\kappa} \bx_{|\kappa(1)|} \bx_{|\kappa(2)|} \dots \bx_{|\kappa(n)|}$$ \index{$\scrF_\bfG$ enriched chromatic function}  where the sum is over all enriched colouring $\kappa$ of $G$. For example, if $\bfG$ is the labelled edge-coloured digraph in \Cref{Fig:labelled-edge-coloured}, then 
\begin{align*}
		\begin{array}{ccccccccc}
			& \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$1$}](d) at (0,5) {$\bffour$};
				\node[squarednode,label=above:{$-1$}](b)  at (-2,2.5) {$\bftwo$};
				\node[squarednode,label=above:{$-1$}](a)  at (0,0) {$\bfone$};
				\node[squarednode, label=above:{$1$}](c)  at (2,2.5) {$\bfthree$};
			\end{tikzpicture} &
			&\begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$1$}](d) at (0,5) {$\bffour$};
				\node[squarednode,label=above:{$1$}](b)  at (-2,2.5) {$\bftwo$};
				\node[squarednode,label=above:{$-1$}](a)  at (0,0) {$\bfone$};
				\node[squarednode, label=above:{$1$}](c)  at (2,2.5) {$\bfthree$};
			\end{tikzpicture} &
			&\begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$2$}](d) at (0,5) {$\bffour$};
				\node[squarednode,label=above:{$-2$}](b)  at (-2,2.5) {$\bftwo$};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {$\bfone$};
				\node[squarednode, label=above:{$1$}](c)  at (2,2.5) {$\bfthree$};
			\end{tikzpicture} & 
			& \begin{tikzpicture}[scale=0.4,
roundnode/.style={circle, draw=black!60, fill=gray!10, very thick, minimum size=5mm},
squarednode/.style={rectangle, draw=black!60, fill=gray!5, very thick, minimum size=5mm},]
				\draw[black,double, thick,->] (a)--(c);
				\draw[black, thick,->] (a)--(b);
				\draw[black, double, thick,->] (b)--(d);
				\node[squarednode,label=above:{$2$}](d) at (0,5) {$\bffour$};
				\node[squarednode,label=above:{$-2$}](b)  at (-2,2.5) {$\bftwo$};
				\node[squarednode,label=above:{$1$}](a)  at (0,0) {$\bfone$};
				\node[squarednode, label=above:{$-2$}](c)  at (2,2.5) {$\bfthree$};
			\end{tikzpicture}\\
			\scrF_G=& \bx_1\bx_1\bx_1\bx_1 & + & \bx_1\bx_1\bx_1\bx_1 & + & \bx_1\bx_2\bx_1\bx_2& + & \bx_1\bx_2\bx_2\bx_2&+\cdots.
		\end{array}
	\end{align*}
\subsection{Peak algebra in noncommuting variables, labelled descent-to-peak map, and new results}\label{subsec:peak-theta-results-n} In this section, we demonstrate that the previous results for the peak algebra and the peak-to-descent map in the literature can be generalized to noncommuting variables. While the results in this section are natural, the proofs are intricate and complex.

The \emph{peak algebra in noncommuting variables}, denoted $\NCPeak$, \index{$\NCPeak$} \index{use , denoted ---, } is the space spanned by
$$\{\scrF_\bfG: \text{$\bfG$ is a labelled edge-coloured digraph}\}.$$
The \emph{labelled descent-to-peak} map $\Theta_{\NCQSym}$ is  \index{\emph{labelled descent-to-peak} map $\Theta_{\NCQSym}$}
 \begin{align}\label{Def:ThetaMapn}
\begin{array}{cccc}
\Theta_\NCQSym:& \NCQSym & \rightarrow & \NCPeak\\
& \scrY_\bfG & \mapsto & \scrF_\bfG.
\end{array} 
\end{align}



\noindent (1) A set $B\subseteq \{2,3,\dots,n-1\}$ is called a \emph{peak set}\footnote{The \emph{peak set of a permutation} $\sigma\in \mathfrak{S}_n$ is the set $\peak(\sigma)=\{i\in \{2,3,\dots,n-1\}: \sigma(i-1) < \sigma(i) $$> \sigma(i+1)\}$. If $i\in \peak(\sigma)$, then $i-1,i+1\not\in \peak(\sigma)$. Thus each peak set is the peak set of a  permutation.} if $b\in B$ implies that $\{b-1,b+1\}\cap B=\emptyset$.  By \cite[Theorem 3.1]{S97} the dimension of the space of the homogeneous elements of degree $n$ of $\Pi$, $\Pi_n$, is $|\{B \subseteq\{2,3,\dots,n-1\} \text{~is a peak set}\}|=|\{\alpha\vDash [n]: \text{$\alpha$ is an odd composition} \}|=f_n,$  the $n$th Fibonacci number. 
%\begin{align*}
%\dim(\Pi_n)&=|\{B \subseteq\{2,3,\dots,n-1\} \text{~is a peak set}\}|\\
%&=|\{\alpha\vDash [n]: \text{$\alpha$ is an odd composition} \}|\\
%&=f_n,
%\end{align*}
%where $f_n$ is the $n$th Fibonacci number. 
 A \emph{set composition} $\phi$ of $[n]$,  \index{\emph{set composition} $\phi$ of $[n]$} denoted $\phi\vDash[n]$, is a sequence of mutually disjoint nonempty sets whose union is $[n]$. An \emph{odd set composition} is a set composition whose all blocks have odd sizes. 
 \begin{thm} The dimension of the space of homogeneous elements of degree $n$ of $\NCPeak$, $\NCPeak_n$, is 
{  
  \setlength{\abovedisplayskip}{.5em}
\begin{align*}
\dim(\NCPeak_n)& = |\{(B,\sigma): B\subseteq \{2,3,\dots,n-1\} \text{~is a peak set and~}\Des(\sigma)\subseteq \odd(B) \}| \\
&=|\{\phi\vDash [n]: \text{$\phi$ is an odd set composition} \}|=a_n
\end{align*}}
where 
$\Des(\sigma)=\{i\in [n-1]: \sigma(i)>\sigma(i+1)\},$ and $a_n$ is the sequence A006154 in the OEIS.\\ 
\end{thm}

\noindent (2) The peak algebra $\Pi$ is a Hopf algebra \cite[Theorem 2.2]{BMSvW}. 
\begin{thm} $\NCPeak$  is a Hopf algebra. \end{thm} 

\noindent (3) The descent-to-peak map, $\Theta_{\QSym}$, is a Hopf algebra morphism \cite[Section 2]{BMSvW}. 
\begin{thm} 
The labelled descent-to-peak map,
 $\Theta_{\NCQSym}$, is a surjective Hopf algebra morphism and the following diagram commutes.
 
\begin{center} \scalebox{.9}{
\begin{tikzpicture}
	\node(NCSym) at (4,0){$\mathsf{\NCQSym}$};
	\node(NCQSym) at (8,0){$\QSym$};
	\node(NCGamma) at (4,-4){$\NCPeak$};
	\node(NCPeak) at (8,-4){$\Pi$};
	
	\draw[thick,->]  (NCSym)->(NCQSym);
	\draw[thick,->]  (NCGamma)->(NCPeak);
	\draw[thick,->]   (NCSym)->(NCGamma);
	\draw[thick,->]   (NCQSym)->(NCPeak);
	
	\node(iota1) at (6,0.2){$\rho$};
	\node(iota2) at (6,-3.8){$\rho$};
	\node(ThetaNCSym) at (3.15,-2){$\Theta_{\NCQSym}$};
	\node(ThetaNCQSym) at (8.65,-2){$\Theta_\QSym$};
	
	\node(yg) at (4.5,-1){$\scrY_\bfG$};
	\node(fg) at (4.5,-3){$\scrF_\bfG$};
	
		\node(xg) at (7.5,-1){$\scrX_G$};
	\node(eg) at (7.5,-3){$\scrE_G$};
	
	
        %\node(yg') at (11.5,-1){$\scrY_{G'}$};
	%\node(fg') at (11.5,-3){$\scrF_{G'}$};
	
	\draw[thick,|->] (yg)--(fg);
	\draw[thick,|->] (xg)--(eg);
	\draw[thick,|->] (yg)--(xg);
	\draw[thick,|->] (fg)--(eg);
	%\draw[thick,|->] (fg)--(fg');
\end{tikzpicture} }
\end{center}
\end{thm} 

\noindent (4) Given a subset $A=\{a_1<a_2<\cdots<a_{k}\}$ of $[n-1]$
 and $\sigma\in\mathfrak{S}_n$, we say $(A,\sigma)$ is \emph{standard} if $\Des(\sigma)\subseteq A.$ The value of the labelled descent-to-peak map at the monomial and fundamental bases elements of $\NCQSym$ are as follows. 
 
(a) {\bf Fundamental basis.} 
Let $A=\{a_1<a_2<\cdots<a_{k}\}$ of $[n-1]$
 and $\sigma\in\mathfrak{S}_n$ such that $(A,\sigma)$ is standard. The \emph{fundamental} basis element $\bF_{(A,\sigma)}$ of $\NCQSym$ is the generalized chromatic function in noncommuting variables of the labelled edge-coloured digraph
 $$\bfG=Q_{\{\sigma(1),\dots,\sigma(a_1)\}}\rightarrow Q_{\{\sigma(a_1+1), \dots, \sigma(a_2)\}}\rightarrow \dots \rightarrow Q_{\{\sigma(a_k+1),\dots,\sigma(n)\}}.$$
 Therefore, by the description of the labelled descent-to-peak map in (\ref{Def:ThetaMapn})
 \begin{thm}
 $ \displaystyle{
 \Theta_{\NCQSym}(\bF_{(A,\sigma)})=\Theta_{\NCQSym}(\scrY_{\bfG})=\scrF_\bfG.}$
 \end{thm}
 
(b) {\bf Monomial basis.} Let $A=\{a_1<a_2<\cdots<a_{k}\}$ of $[n-1]$
 and $\sigma\in\mathfrak{S}_n$ such that $(A,\sigma)$ is standard. The \emph{monomial} basis element $\bM_{(A,\sigma)}$ of $\NCQSym$ is
$$\bM_{(A,\sigma)}=\sum_{C\subseteq A} (-1)^{|A|-|C|} \bF_{(C,\sigma)}.$$
For each peak set $B\subseteq \{2,3,\dots, n-1\}$, the \emph{monomial peak function in noncommuting variables} $\boldeta_{(B,\sigma)}$ is 
$$\boldeta_{(B,\sigma)}=(-1)^{|B|}\sum_{A\subseteq \odd(B)} 2^{|A|+1}\bM_{(A,\sigma)}.$$
\begin{thm}
We have
\begin{align*}
\Theta_{\QSym}(\bM_{(A,\sigma)})
&=\begin{cases}
(-1)^{n-1-|B|-|A|} \boldeta_{(B,\sigma)} & \text{if $n-\max(A)$ is odd,}\\
0 & \text{otherwise.}
\end{cases} 
\end{align*}
\end{thm}

\noindent (5)   The Hopf algebra of \emph{Schur's Q-functions} introduced in \cite{S11} is denoted by $\Gamma$ and by \cite[Theorem 3.8]{S97}, $\Gamma$ is the intersection of the Peak algebra $\Pi$ and the Hopf algebra of symmetric functions $\Sym$, that is, 
$\Gamma=\Pi\cap \Sym.$
Moreover, by \cite[Theorem 3.1]{S97}
$\Theta_{\QSym}(\Sym)=\Gamma.$

We define the Hopf algebra of \emph{Schur's Q-functions in noncommuting variables}, denoted $\NCGamma$,  to be the intersection of the peak algebra in noncommuting variables and the Hopf algebra of symmetric functions in noncommuting variables,  
$ \displaystyle{
\NCGamma=\NCPeak\cap \NCSym.
}$
\begin{thm} 
We have
$\Theta_\NCQSym(\NCSym)=\NCGamma.$
\end{thm} 



\noindent (6) The restriction of the descent-to-peak map $\Theta_{\QSym}$ to $\Sym$ is denoted by $\Theta_\Sym$.  The restriction of the labelled descent-to-peak map $\Theta_{\NCQSym}$ to $\NCSym$ is denoted by $\Theta_\NCSym$. 

\begin{thm} The following diagram commutes.
\begin{center} \scalebox{.87}{
\begin{tikzpicture}
	\node(Sym) at (5,0){${\Sym}$};
	\node(QSym) at (8,0){$\QSym$};
	\node(Gamma) at (5,-4){$\Gamma$};
	\node(Pi) at (8,-4){$\Pi$};
	
	\node(NCSym) at (3,2){$\NCSym$};
	\node(NCQSym) at (11,2){$\NCQSym$};
	\node(NCGamma) at (3,-6){$\NCGamma$};
	\node(NCPeak) at (11,-6){$\NCPeak$};

	\draw[thick,->]  (Sym)->(QSym);
	\draw[thick,->]  (Gamma)->(Pi);
	\draw[thick,->]   (Sym)->(Gamma);
	\draw[thick,->]   (QSym)->(Pi);
	
	\draw[thick,->]  (NCSym)->(NCQSym);
	\draw[thick,->]  (NCGamma)->(NCPeak);
	\draw[thick,->]   (NCSym)->(NCGamma);
	\draw[thick,->]   (NCQSym)->(NCPeak);
	
	
	\node(iota1) at (6.5,0.2){$\iota$};
	\node(iota2) at (6.5,-3.8){$\iota$};
	\node(ThetaSym) at (5-0.5,-2){$\Theta_{\Sym}$};
	\node(ThetaQSym) at (7.3,-2){$\Theta_\QSym$};
	
        \node(iota3) at (6.5,2.2){$\iota$};
	\node(iota4) at (6.5,-3.8-2){$\iota$};
	\node(ThetaNCSym) at (2.3,-2){$\Theta_{\NCSym}$};
	\node(ThetaNCQSym) at (11.9,-2){$\Theta_\NCQSym$};
	
	\node(xg) at (8.7,-0.7){$\scrX_G$};
	\node(eg) at (8.7,-3.3){$\scrE_G$};
	\draw[thick,|->] (xg)--(eg);
	
	\node(yg) at (10.4,0.5){$\scrY_\bfG$};
	\node(fg) at (10.4,-4.5){$\scrF_\bfG$};
	
	\draw[thick,|->] (yg)--(fg);
	\draw[thick,|->] (yg)--(xg);
	\draw[thick,|->] (fg)--(eg);
	
	\draw[thick,->] (NCQSym)--(QSym);
	\draw[thick,->] (NCPeak)--(Pi);
	\draw[thick,->] (NCGamma)--(Gamma);
	\draw[thick,->] (NCSym)--(Sym);

\end{tikzpicture} }
\end{center}
\end{thm}

\noindent (7) A set partition $\pi=\pi_1/\pi_2/\cdots/\pi_{l}$ of $[n]$, denoted $\pi=\pi_1/\pi_2/\cdots/\pi_{l}\vdash [n]$, is the set of mutually disjoint non-empty subsets $\pi_1,\pi_2,\dots,\pi_{l}$ of $[n]$ whose their union is $[n]$. 
\index{$\pi$ set partition}

 Given a labelled edge-coloured digraph $\bfG$ with vertex set $S$ and $\sigma\in \mathfrak{S}_S$, define $\sigma\circ \bfG$ to be the labelled edge-coloured digraph with vertex set $S$ in which 
\begin{itemize}[itemsep=0pt,topsep=.5em]
 \item $i\Rightarrow j$ in $\bfG$ if and only if ${\sigma(i)}\Rightarrow {\sigma(j)}$ in $\sigma\circ \bfG$.
\item $i\rightarrow j$ in $\bfG$ if and only if ${\sigma(i)}\rightarrow {\sigma(j)}$ in $\sigma\circ \bfG$.
\end{itemize} 



 The values of the labelled descent-to-peak map at different bases of $\NCSym$ are as follows.
 
 (a)  For $\pi=\pi_1/\pi_2/\cdots/\pi_{l}\vdash [n]$, we have
$$\Theta_\NCSym(\bfp_\pi)=\begin{cases}
2^{l}\bfp_\pi & \text{if all blocks of $\pi$ have odd sizes,}\\
0 & \text{otherwise.}
\end{cases} $$

(b)  For $\pi=\pi_1/\pi_2/\cdots/\pi_{l}\vdash [n]$, by \cite[Section 10]{ALvW22},  we have that the \emph{complete homogeneous symmetric function in noncommuting variables} $\bfh_\pi$ (resp.,  \emph{elementary symmetric function in noncommuting variables} $\bfe_\pi$) is 
$$\bfh_\pi=\sum_{\sigma\in \frakS_{\pi}} \scrY_{\sigma\circ Q_{\pi}},~~~{\text{(resp.}}~ \bfe_\pi=\sum_{\sigma\in \frakS_{\pi}} \scrY_{\sigma\circ P_{\pi}}),\quad \text{where}$$   
\begin{center} $Q_\pi=Q_{\pi_1}\sqcup Q_{\pi_2}\sqcup \cdots \sqcup Q_{\pi_{\ell(\pi)}},$ (resp. $P_\pi=P_{\pi_1}\sqcup P_{\pi_2}\sqcup \cdots \sqcup P_{\pi_{\ell(\pi)}}$)
and $\mathfrak{S}_{\pi}=\mathfrak{S}_{\pi_1}\times \mathfrak{S}_{\pi_2} \times \cdots \times \mathfrak{S}_{\pi_{\ell(\pi)}}.$
\end{center} 
Thus by (\ref{Def:ThetaMapn}) and the fact that $\scrF_{\sigma\circ P_{\pi}}=\scrF_{\sigma\circ Q_{\pi}}$, 
$$\Theta_{\NCSym}(\bfh_\pi)=\sum_{\sigma\in \frakS_{\pi}} \scrF_{\sigma\circ Q_{\pi}}=\Theta(\bfe_\pi)~~~\text{(resp.}~ \Theta_{\NCSym}(\bfe_\pi)=\sum_{\sigma\in \frakS_{\pi}} \scrF_{\sigma\circ P_{\pi}}=\Theta(\bfh_\pi)).$$ 


(c) For $\pi=\pi_1/\pi_2/\cdots/\pi_{l}\vdash [n]$, define the permutation $\delta_\pi\in\mathfrak{S}_n$ such that  
if $\delta_\pi(i)\in \pi_s$ and $\delta_\pi(i+1)\in \pi_t$, then either $s=t$ and $\delta_\pi(i)<\delta_\pi(i+1)$ or $\min(\pi_s)<\min(\pi_t)$. For example, if $\pi=156/24/36$, then $\delta_\pi=1562436$.
The \emph{Schur function in noncommuting variable} $\bfs_\pi$ is 
 \[\bfs_\pi=\delta_{\pi}\circ \bfdet \left( \scrY_{Q_{[\lambda_i-i+j]}}  \right)_{1\leq i,j\leq l}.\]
 Thus by (\ref{Def:ThetaMapn}),
$$\Theta_{\NCSym}(\bfs_\pi)=\delta_{\pi}\circ \bfdet \left( \scrF_{Q_{[\lambda_i-i+j]}}  \right)_{1\leq i,j\leq l}.$$




\noindent (8) In \cite{S11} Schocker proves that the peak algebra is a left co-ideal of $\QSym$ under internal comultiplication, a generalized dual Kronecker product.  We provide a combinatorial proof for the following theorem. 
\begin{thm} The peak algebra in noncommuting variables is a left co-ideal of $\NCQSym$ under internal comultiplication.
\end{thm}
\acknowledgements{The authors would like to thank Nantel Bergeron for introducing the problem and to Stephanie van Willigenburg for helpful conversations. 
Both authors were supported in part by the Fundamental Research Fund for the Central Universities and National Nature Science Foundation of China (NSFC), Project No. 12301421.}

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