text:
\begin{document}

\begin{abstract}
The Euler number $E_n$ (resp. Entringer number $E_{n,k}$)
enumerates the alternating (down-up) permutations of
 $\{1,\dots,n\}$ (resp.  starting with $k$). The Springer number $S_n$
 (resp. Arnold number $S_{n,k}$)
enumerates the type $B$ alternating permutations
 (resp.  starting with $k$).
In  this paper, using bijections
 we first derive the counterparts
in {\em Andr\'e permutations} and {\em Simsun  permutations}
for the Entringer numbers $(E_{n,k})$, and then  the counterparts
in {\em signed Andr\'e permutations} and {\em type $B$ increasing 1-2
trees} for the Arnold numbers $(S_{n,k})$.
\end{abstract}

\maketitle

\tableofcontents
\section{Introduction}

The \emph{Euler numbers} $E_n$ are defined by
the exponential generating function
$$
1 + \sum_{n\ge 1} E_n \frac{x^n}{n!} = \tan x + \sec x.
$$
This is the sequence A000111 in \cite{Slo18}.
In 1877 Seidel \cite{Sei77} defined the triangular array $(E_{n,k})$ by
the recurrence
\begin{align}
E_{n,k} = E_{n,k-1} + E_{n-1,n+1-k} \qquad (n \ge k\ge 2)
\label{eq:rec}
\end{align}
with $E_{1,1}=1$, $E_{n,1}=0$ ($n\ge2$), and proved that $E_n = \sum_k E_{n,k}$, i.e., the Euler number $E_n$ is the sum of the entries of the $n$-th row of the following triangle:

\begin{align}\label{tab:kempner}
\begin{tabular}{ccccccccc}
& &          &&  $E_{1,1}$ \\
&& & $E_{2,1}$ & $\rightarrow$ & $E_{2,2}$ \\
&& $E_{3,3}$ & $\leftarrow$ & $E_{3,2}$ & $\leftarrow$ & $E_{3,1}$ \\
&$E_{4,1}$ & $\rightarrow$ & $E_{4,2}$ & $\rightarrow$ & $E_{4,3}$ & $\rightarrow$ & $E_{4,4}$& \\

& & & & $\cdots$
\end{tabular}
=

\begin{tabular}{ccccccccc}
&&&          &  1 \\
&& & 0 & $\rightarrow$ & 1 \\
&& 1& $\leftarrow$ & 1 & $\leftarrow$ &0  \\
&0 & $\rightarrow$ & 1& $\rightarrow$ & 2& $\rightarrow$ & 2 \\

& & & & $\cdots$ \end{tabular}
\end{align}
Ths first few values of
$\E_{n,k}$ are given in Table~\ref{tab:EntA}.

\begin{table}[t]
\centering
\begin{tabular}{c|ccccccc|c}
$n\setminus k$ & 1 & 2 & 3 & 4 & 5 & 6 & 7&$E_n$ \\ \hline
1 & 1&&&&&&&1 \\
2 & 0 & 1&&&&&& 1\\
3 & 0 & 1 & 1&&&&&2 \\
4 & 0 & 1 & 2 & 2&&&&5 \\
5 & 0 & 2 & 4 & 5 & 5&&&16\\
6 & 0 & 5 & 10 & 14 & 16 & 16&&61\\
7 & 0 & 16 & 32 & 46 & 56 & 61 & 61&271\\
\end{tabular}
\vspace{1em}
\caption{The Entringer numbers $\Enk$ for $1\leq k \leq n\leq 7$
and Euler numbers $\En=\sum_{k=1}^n \Enk$.}
\label{tab:EntA}
\end{table}

Andr\'e~\cite{And79} showed in  1879 that the Euler number $E_n$ enumerates
the alternating permutations of $[n]:=\set{1,2,\dots,n}$, i.e., the permutations
$\sigma_1\sigma_2\dots\sigma_n$ of $12\ldots n$ such that $\sigma_1 > \sigma_2 < \sigma_3 > \sigma_4 < \cdots.$
Let $\DUn$ be the set of (down-up) alternating permutations

of $[n]$.
For example,
$$
\DU_4=\{2143, 3142, 3241, 4132, 4231\}.
$$
In 1933
Kempener~\cite{Kem33} used the boustrophedon algorithm~\eqref{tab:kempner} to enumerate  alternating permutations without refering to Euler numbers. Since  Entringer~\cite{Ent66} first
found the combinatorial interpretation of
Kempener's table $(E_{n,k})$  in terms of  Andr\'e's  model for Euler numbers,
the numbers $E_{n,k}$ are then called \emph{Entringer numbers}.
\begin{thm}[Entringer]
The number of the (down-up) alternating permutations of $[n]$ with first entry $k$ is $E_{n,k}$,
i.e., $E_{n,k} = \# \DUnk$, where
$$
\DUnk := \set{\sigma\in\DUn : \sigma_1=k}.
$$
\end{thm}

According to Foata-Sch\"utzenberger~\cite{FS71}

a sequence of sets $(X_{n})$ is called an \emph{Andr\'e complex}
if the cardinality of $X_{n}$ is equal to $E_{n}$ for $n\geq 1$.
Several other  Andr\'e complexs were introduced also in \cite{FS71} such as \emph{Andr\'e permutations of first and second kinds}, \emph{Andr\'e trees} or \emph{increasing 1-2 trees}
and
Rodica Simion and Sheila Sundaram~\cite{Sta10} discovered the \emph{Simsun} permutations; see Section~2.
A sequence of sets $(X_{n,k})$ is called an \emph{Entringer family}
if the cardinality of $X_{n,k}$ is equal to $E_{n,k}$ for $1 \le k \le n$.
During the last two decades of the twentieth century,
Poupard worked out several Entringer families in a series of papers~\cite{Pou82, Pou89, Pou97}.
Stanley \cite[Conjecture 3.1]{Sta94} and Hetyei \cite{Het96} introduced more Entringer families
by refining of Purtill's result~\cite[Theorem 6.1]{Pur93} about the $cd$-index of Andr\'e and Simsun permutations with fixed last letter.

The \emph{Springer numbers} $S_n$ are
defined by the exponential generating function~\cite{Spr71}
$$
1 + \sum_{n\ge 1} S_n \frac{x^n}{n!}
= \frac{1}{\cos x - \sin x}.

$$
Arnold~\cite[p.11]{Arn92} showed in 1992 that $S_n$ enumerates a signed-permutation analogue of the alternating permutations.
Recall that a \emph{signed permutation} of $[n]$ is a sequence $\pi=(\pi_1, \ldots, \pi_n)$ of elements of $[\pm n]:=\{-1,\dots, -n\}\cup \{1,\dots,n\}$ such that $|\pi|=(|\pi_1|, \ldots, |\pi_n|)$ is a permutation of $[n]$.
We write $\mathcal{B}_n$ for  the set of all signed permutations of $[n]$.
Clearly the cardinality of $\mathcal{B}_n$ is $2^n n!$.
An \emph{(down-up) alternating permutation of type $B_n$}

is a signed permutation  $\pi\in \mathcal{B}_n$ such that $\pi_1>\pi_2<\pi_3>\pi_4\cdots$
and  a \emph{snake} of type $B_n$ is an alternating permutation of type $B_n$ starting with a positive entry.
Let $\DUn^{(B)}$ be the set of (down-up) alternating permutations of type $B_n$
and $\DUnB$ the set of snakes of type $B_n$.
Clearly the cardinality of $\DUn^{(B)}$ is $2^n E_n$.
Arnold~\cite{Arn92} showed that the Springer number $S_n$ enumerates the snakes of type $B_n$.
For example,
the $11$ snakes of $\DUB_3$ are  as follows:
$$
1\,\bar{2}\,3,
~1\,\bar{3}\,2,
~1\,\bar{3}\,\bar{2},
~2\,1\,3,
~2\,\bar{1}\,3,
~2\,\bar{3}\,1,
~2\,\bar{3}\,\bar{1},
~3\,1\,2,
~3\,\bar{1}\,2,
~3\,\bar{2}\,1,
~3\,\bar{2}\,\bar{1},
$$
where we write $\bar{k}$ for $-k$. Arnold~\cite{Arn92} introduced the following
 pair of triangles to compute the Springer numbers:
\begin{align*}
\begin{tabular}{ccccccccc}
&&&&	\scriptsize $S_{1,-1}$ \\
&&&	\scriptsize $S_{2,2}$ & $\leftarrow$  & \scriptsize $S_{2,1}$ \\
&&	\scriptsize $S_{3,-3}$ & $\rightarrow$ & \scriptsize $S_{3,-2}$ & $\rightarrow$ & \scriptsize $S_{3,-1}$  \\
&	\scriptsize $S_{4,4}$ & $\leftarrow$ & \scriptsize $S_{4,3}$ & $\leftarrow$  & \scriptsize $S_{4,2}$ &
$\leftarrow$  & \scriptsize $S_{4,1}$ \\

& & & & $\cdots$\\
\\
& & & & $\Updownarrow$\\
\\
&&&&	1 \\
&&&	2 & $\leftarrow$  & 1 \\
&&	0 & $\rightarrow$ & 2 & $\rightarrow$ & 3  \\
&	16 & $\leftarrow$ & 16 & $\leftarrow$  & 14 & $\leftarrow$  & 11 \\

& & & & $\cdots$
\end{tabular}
&&&
\begin{tabular}{ccccccccc}
&&&& 	\scriptsize $S_{1,1}$ \\
&&&	\scriptsize $S_{2,-1}$ & $\leftarrow$  & \scriptsize $S_{2,-2}$ \\
&&	\scriptsize $S_{3,1}$ & $\rightarrow$ & \scriptsize $S_{3,2}$ & $\rightarrow$ & \scriptsize $S_{3,3}$  \\
&	\scriptsize $S_{4,-1}$ & $\leftarrow$ & \scriptsize $S_{4,-2}$ & $\leftarrow$  & \scriptsize $S_{4,-3}$ & $\leftarrow$  & \scriptsize $S_{4,-4}$ \\

& & & & $\cdots$\\
\\
& & & & $\Updownarrow$\\
\\
&&&&	1 \\
&&&	1 & $\leftarrow$  & 0 \\
&&	3 & $\rightarrow$ & 4 & $\rightarrow$ & 4  \\
&	11 & $\leftarrow$ & 8 & $\leftarrow$  & 4 & $\leftarrow$  & 0 \\

& & & & $\cdots$
\end{tabular}
\end{align*}
where $S_{n,k}$ is defined by
$S_{1,1}=S_{1,-1}=1$, $S_{n,-n}=0$ ($n \ge 2$), and the recurrence
\begin{align}
S_{n,k} &=
\begin{cases}
S_{n,k-1}+ S_{n-1,-k+1} & \text{if $n \ge k >1$,}\\
S_{n,-1} & \text{if $n>k=1$,}\\
S_{n,k-1} + S_{n-1,-k}& \text{if $-1 \ge k > -n$.}
\end{cases}
\label{eq:recB}
\end{align}
\begin{thm}[Arnold]
For all integers $1\le k\le n$,
the number of the snakes of type $B_n$ starting with $k$ is $S_{n,k}$,
i.e., $S_{n,k} = \# \DUnkB$ with  $S_n = \sum_{k>0} \#\DUnkB$, where
$$
\DUnkB := \set{\sigma\in\DUnB : \sigma_1=k}.
$$

Moreover, for all integers $-n \le k \le -1$, it holds that
\begin{align*}
S_{n,k} = \#\set{\sigma\in \DUn^{(B)} : \sigma_1=k }.
\end{align*}
\end{thm}

Similarly, the numbers $S_{n,k}$ are called \emph{Arnold numbers} and
a sequence of sets $(X_{n,k})$ is called an \emph{Arnold family} if the  cardinality of $X_{n,k}$  is equal to $S_{n,k}$ for $1 \le \abs{k} \le n$.
The first values of Arnold and Springer numbers are given in Table~\ref{tab:Arnold}.
\begin{table}[t]
\centering
\begin{tabular}{c|cccccc|cccccc|c}
$n\setminus k$ & -6 & -5 & -4 & -3 & -2 & -1  & 1 & 2 & 3 & 4 & 5 & 6 & $S_n$ \\
\hline
1 &     &     &     &     &     & 1   & 1   &     &     &     &     &     & $1$    \\
2 &     &     &     &     & 0   & 1   & 1   & 2   &     &     &     &     & $3$    \\
3 &     &     &     & 0   & 2   & 3   & 3   & 4   & 4   &     &     &     & $11$   \\
4 &     &     & 0   & 4   & 8   & 11  & 11  & 14  & 16  & 16  &     &     & $57$   \\
5 &     & 0   & 16  & 32  & 46  & 57  & 57  & 68  & 76  & 80  & 80  &     & $361$  \\
6 & 0   & 80  & 160 & 236 & 304 & 361 & 361 & 418 & 464 & 496 & 512 & 512 & $2763$
\end{tabular}
\vspace{1em}
\caption{The Arnold numbers $\EnkB$ for $1\leq \abs{k} \leq n\leq 6$ and Springer numbers $\EnB=\sum _{k=1}^n\EnkB$.}
\label{tab:Arnold}
\end{table}
The aim of this paper is to provide some new  Entringer families and
new  Arnold families by
refining  known  combinatorial models for  Euler and Springer numbers.
To this end we shall build
bijections between these new  Entringer (resp. Arnold) families with the known
ones.
We refer the reader to the more recent papers~\cite{Sta10, GSZ11, CEFP12, JNT12, FH14} related to the combinatorics of Euler numbers and Springer numbers.

This paper is organized as follows.
In Section~\ref{sec:defn}, we shall give the necessary definitions
 and present our main results. The proof of
our  theorems will be given in Sections~\ref{sec:proof1}-\ref{sec:proof2}.
In Section~\ref{sec:psi},
we shall give more insightful description of two important  bijections.
More precisely, Chuang et al.'s constructed a
$\phi: \T_{n+1} \to \RS_{n}$ in \cite{CEFP12}, we show that
 $\phi$ can be factorized as the compositions of two of our simpler bijections, and
 then
  give a  direct description of
Gelineau et al.'s bijection  $\psi: \DUn \to \Tn$ in \cite{GSZ11}.

\section{Definitions and main results} \label{sec:defn}

Let  $V$ be a finite subset of $\N$.
An \emph{increasing 1-2 tree} on $V$ is a vertex labeled rooted tree
with at most two (upward) branchings at any vertex and vertex labels in increasing order on any (upward) path from the root, see Figure~\ref{fig1}.
In what follows, when one draws an increasing 1-2 tree,
let's designate the \emph{left} child
if its parent has a unique child or it is smaller than the other sibling
and the others, except of the root, are designate the \emph{right} child.

\begin{figure}[t]\label{fig1}

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\caption{increasing 1-2 trees on $[4]$}
\end{figure}

For each vertex $v$ of a binary tree, by exchanging the left and right subtrees of $v$,
we obtain another binary tree. This operation  is called a \emph{flip}.
If two binary trees can be connected by a sequence of flips,
we say that these two trees are \emph{flip equivalent}.
Since flip equivalence is obviously an equivalence relation,
we are able to define the equivalence classes of binary trees,
which are corresponding to increasing 1-2 trees;  see \cite[Section 3.2]{Sta10} in detail.

\begin{defn}
Given an increasing 1-2 tree $T$,
the {\em minimal path} of $T$ is the unique sequence $(v_1, \cdots, v_{\ell})$ of vertices where $v_1$ is the root,
$v_{k+1}$ is the left child of $v_k$ ($1\le k<\ell$), and $v_\ell$ is a leaf.
The vertex $v_\ell$ is called the \emph{minimal leaf} of $T$ and denoted by  $\Pleaf(T)$. Similarly, the unique path $(v_1, \cdots, v_{\ell})$ from $v_1=v$
 to a leaf $v_{\ell}$ of $T$
is called the \emph{maximal path} from $v$ if
 $v_{k+1}$ is the right child of $v_k$ for $1\le k<\ell$.
\end{defn}
Let $\T_V$ be the set of increasing 1-2 trees on $V$ with
$\T_{n}: = \T_{[n]}$ and
$$
\T_{n,k} = \set{T\in\T_n : \Pleaf(T)=k}.
$$

Donaghey~\cite{Don75} (see also \cite{Cal05}) proved bijectively that the Euler number $E_n$ enumerates
the binary increasing trees in $\T_n$ and
Poupard~\cite{Pou82} showed that
the sequence $(\T_{n,k})$ is an Entringer family.
In a previous work Gelineau et al.~\cite{GSZ11} proved bijectively Poupard's result by
establishing a
bijection between  $\DUn$ and $\Tn$.
\begin{thm}[Gelineau-Shin-Zeng]
\label{thm:GSZ11}
There is an explicit  bijection $\psi: \DUn \to \Tn$ such that
$$\Pleaf(\psi(\sigma))=\First(\sigma)$$
 for all $\sigma\in \DUn$, where
$\First(\sigma)$  is the first entry of $\sigma$.
\end{thm}

Let $\S_n$ be the group of permutations on $[n]$.
For a permutation $\sigma = \sigma_1 \dots \sigma_n \in \S_n$,
a \emph{descent} (resp. \emph{ascent}) of $\sigma$ is a pair $(\sigma_i, \sigma_{i+1})$
with $\sigma_i > \sigma_{i+1}$ (resp. $\sigma_i < \sigma_{i+1}$) and $1 \le i \le n-1$,
a \emph{double descent} of $\sigma$ is a triple $(\sigma_i, \sigma_{i+1}, \sigma_{i+2})$ with $\sigma_i > \sigma_{i+1} > \sigma_{i+2}$ and $1 \le i \le n-2$, and
a \emph{valley} of $\sigma$ is a triple $(\sigma_i, \sigma_{i+1}, \sigma_{i+2})$ with
$\sigma_i > \sigma_{i+1} < \sigma_{i+2}$ and $1 \le i \le n-2$.

Hetyei \cite[Definition 4]{Het96} defined recursively \emph{Andr\'e permutation of second kind}
if it is empty or satisfies the following:
\begin{enumerate}[(i)]
\item $\sigma$ has no double descents.
\item $(\sigma_{n-1}, \sigma_n)$ is not descent, i.e., $\sigma_{n-1} < \sigma_n$.
\item For all $2\le i \le n-1$, if $(\sigma_{i-1}, \sigma_{i}, \sigma_{i+1})$ is a valley of $\sigma$, then the minimum letter of $w_2$ is larger than the minimum letter of $w_4$ for the $\sigma_i$-factorization $(w_1, w_2, \sigma_i, w_4, w_5)$ of $\sigma$,
where the word $w_1 w_2 \sigma_i w_4 w_5$ is equal to $\sigma$
and $w_2$ and $w_4$ are maximal consecutive subwords of $\sigma$ satisfies its all letters are greater than $\sigma_i$.
\end{enumerate}

It is known that the above definition for Andr\'e permutation of second kind is simply equivalent to the following definition. Let  $\sigma_{[k]}$ denote the \emph{subword} of $\sigma$ consisting of $1,\dots, k$ in the order they appear in $\sigma$.
\begin{defn}
\label{def:andre}
A permutation $\sigma \in \S_n$ is called an \emph{Andr\'e permutation}
if $\sigma_{[k]}$ has no double descents and ends with an ascent for all $1\leq k\leq n$.
\end{defn}
For example, the permutation $\sigma = 43512$ is not Andr\'e
since the subword $\sigma_{[4]}=4312$ contains a double descent $(4,3,1)$,
while the permutation $\tau = 31245$ is Andr\'e since there is no double descent in the subwords:
\begin{align*}
\tau_{[1]} &= 1, &
\tau_{[2]} &= 12, &
\tau_{[3]} &= 312, &
\tau_{[4]} &= 3124, &
\tau_{[5]} &= 31245.
\end{align*}
Foata and Sch{\"u}tzenberger~\cite{FS73} proved that the Euler number $E_n$ enumerates the
Andr\'e permutations in $\S_n$.
Let $\DAn$ be the set of Andr\'e permutations in $\S_n$.
For example, $$\DA_{4} = \set{1234, 1423, 3124, 3412, 4123}.$$

\begin{rmk}
\label{rem:andre}
Foata and Sch{\"u}tzenberger in \cite{FS73} introduced \emph{augmented Andr\'e permutation}
is a permutation $\sigma \in \S_n$,
if $\sigma$ has no double descents, $\sigma_n = n$, and, for $1<j<k\le n$ satisfying
\begin{align*}
\sigma_{j-1} &= \max\set{\sigma_{j-1}, \sigma_{j}, \sigma_{k-1}, \sigma_k} && \text{and}&
\sigma_k     &= \min\set{\sigma_{j-1}, \sigma_{j}, \sigma_{k-1}, \sigma_k},
\end{align*}
there exists $\ell$ such that $j<\ell <k$ and $\sigma_\ell < \sigma_k$.
\end{rmk}

\begin{defn}
\label{def:simsun}
A permutation $\sigma \in \S_n$ is called a \emph{Simsun permutation}
if $\sigma_{[k]}$ has no double descents for all $1\leq k\leq n$.
\end{defn}
By definition, an Andr\'e permutations is always a Simsun permutation, but the reverse is not true.

For example, the permutation $\sigma=25134$ is Simsun but not Andr\'e, because $\tau_{[2]} = 21$ ends with an descent:
\begin{align*}
\tau_{[1]} &= 1, &
\tau_{[2]} &= 21, &
\tau_{[3]} &= 213, &
\tau_{[4]} &= 2134, &
\tau_{[5]} &= 25134.
\end{align*}
Let $\RSn$ be the set of Simsun permutations in $\S_n$.
For example, $$\RS_{3} = \set{123, 132, 213, 231, 312}.$$

As for $\Ank$, we define two similar refinements of Andr\'e permutations and Simsun permutations as
\begin{align*}
\DAnk &:= \set{\sigma\in\DAn : \sigma_n=k}, &
\RSnk &:= \set{\sigma\in\RSn : \sigma_{n}=k}.
\end{align*}
Some examples are shown in Table~\ref{table:DARS}.

\begin{table}[t]
\centering
\begin{tabular}{c||c|c|c}
$k$ & $\DU_{4,k}$        & $\DA_{4,k}$        & $\RS_{3,k-1}$ \\ \hline
$2$ & $\set{2143}$       & $\set{3412}$       & $\set{231}$ \\
$3$ & $\set{3142, 3241}$ & $\set{1423, 4123}$ & $\set{132, 312}$ \\
$4$ & $\set{4132, 4231}$ & $\set{1234, 3124}$ & $\set{123, 213}$
\end{tabular}
\vspace{1em}
\caption{The sets $\DU_{4,k}$, $\DA_{4,k}$, and $\RS_{3,k-1}$ for $2\le k \le 4$}
\label{table:DARS}
\end{table}

Foata and Han \cite[Theorem 1.1 (iii)]{FH14} proved that  $\DAnk$ is an Entringer family by constructing  a
 bijection between $\DUnk$ and $\DAnk$.
We shall give an easier proof of their result
by  constructing  a simpler bijection $\omega$ between $\Tnk$ and $\DAnk$.
 Of course, combining  $\psi$ (cf. Theorem~\ref{thm:GSZ11}) and $\omega$ we obtain another bijection
between $\DUnk$ and $\DAnk$.
\begin{thm}
\label{main2}
For positive integer $n\ge 1$,
there is a bijection $\omega: \Tn \to \DAn$ such that
\begin{align}
\Pleaf(T)=\Last(\omega(T)) \label{eq:Andre}
\end{align}
for all $T \in \Tn$,
where $\Last(\sigma)$  is the last entry of $\sigma$.
In words, for all $1\le k \le n$,
the mapping $\omega$ is a bijection from $\Tnk$ onto $\DAnk$.
\end{thm}

Whereas one can easily show that the cardinality $\#\A_{n,k}$ of (down-up) alternating permutations of length $n$ with first entry $k$ satisfies \eqref{eq:rec},
it seems hard to show that
the cardinality $\#\DAnk$ of Andr\'e permutations of length $n$ with last entry $k$ or
the cardinality $\#\RSnkone$ of Simsun permutations of length $n-1$ with last entry $k-1$ does.
Thus, in order to show \eqref{eq:Andre} and \eqref{eq:Simsun}, we shall construct a bijection between $\DUnk$, $\DAnk$, $\RSnkone$, and other known Entringer families in \cite{GSZ11}.

Stanley \cite[Conjecture 3.1]{Sta94} conjectured a refinement of Purtill's result~\cite[Theorem 6.1]{Pur93} about the $cd$-index of Andr\'e and Simsun permutations with fixed last letter.
In this conjecture, he mentioned three kinds of Andr\'e permutations:
(i) Augmented Andr\'e permutations in Remark~\ref{rem:andre},

(ii) Andr\'e permutations in Definition~\ref{def:andre}, and

(iii) augmented Sundaram permutations,

where the third corresponds to Simsun permutations in Definition~\ref{def:simsun} by removing last letter.
Hetyei \cite{Het96} proved the conjecture for the second and the third by verifying that both sides satisfy the same recurrence.
In particular, he proves the following result.
\begin{thm}[Hetyei]

For all $1\le k \le n$, we have that two cardinalities of $\DAnk$ and $\RSnkone$ are same, that is,
\begin{align}
\# \DAnk &= \# \RSnkone.
\end{align}
\end{thm}

In the next theorem, we give a bijective
proof of  the conjecture of Stanley by constructing an explicit  bijection.
\begin{thm}\label{main1}
For positive integer $n\ge 1$, there is
a bijection $\varphi: \DAn\to \RSnone$ such that
\begin{align}
\Last(\sigma)-1=\Last(\varphi(\sigma))  \label{eq:Simsun}
\end{align}
for all $\sigma \in \DAn$.
In words,  the mapping  $\varphi$ is a bijection  from $\DAnk$ onto
$\RSnkone$.
Moveover, the bijection $\varphi$ preserves the $cd$-index of Andr\'e and Simsun permutation.
\end{thm}

Given a permutation $\sigma \in B_n$,

denote $\sigma_{[k]}$ the \emph{subword} of $\sigma$ consisting of $k$ smallest entries in the order they appear in $\sigma$.
A \emph{signed Andr\'e permutation} of $[n]$ is a permutation $\sigma \in B_n$
such that $\sigma_{[k]}$ has no double descents and ends with an ascent for all $1\leq k\leq n$.
Let $\DAnB$ be the set of signed Andr\'e permutations of $[n]$
and $\DAnkB$ be the set of signed Andr\'e permutations $\sigma$ in $\DAnB$ ending with entry $k$.
For example, the permutation $\sigma = 2\bar{4}\bar{1}35$ is And\'re due to
\begin{align*}
\sigma_{[1]} &= \bar{4}, &
\sigma_{[2]} &= \bar{4}\bar{1}, &
\sigma_{[3]} &= 2\bar{4}\bar{1}, &
\sigma_{[4]} &= 2\bar{4}\bar{1}3, &
\sigma_{[5]} &= 2\bar{4}\bar{1}35.
\end{align*}
Some examples of $\DAB_{3,k}$ are shown in Table~\ref{table:DARSB}.

\begin{defn}
A \emph{type $B$ increasing 1-2 tree} on $[n]$ is a binary tree with $n$ signed labels
in $\{\pm1,\pm2,\dots,\pm n\}$
such that the absolute values of signed labels are distinct and any vertex is greater than its children.
\end{defn}

For example, all type $B$ increasing 1-2 trees on $[3]$ are given in Figure~\ref{fig2}.
\begin{figure}[t]\label{fig2}

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\caption{Sixteen type $B$ increasing 1-2 trees on $[3]$}
\end{figure}
Let $\TnB$ be the set of type $B$ increasing 1-2 trees on $n$ vertices and $\TnkB$ be the set of trees $T$ in $\TnB$ with leaf $k$ as the end of minimal path.
Clearly we have $\TnB = \bigcup_{\abs{k}>0} \TnkB$.

Our second aim is to show that these two refinements are new Arnold families.
Recall that the sequence $\DUnkB$ is an Arnold family for $1\leq \abs{k} \leq n$ as
$$\DUnkB := \set{\sigma\in \DUn^{(B)} : \sigma_1=k }.$$
\begin{thm}\label{thm:main2}
For all $1\le \abs{k} \le n$, there are two bijections
\begin{align}
\psi^B&: \DUnkB \to \TnkB, \label{eq:TreeB}\\
\omega^B&: \TnkB \to \DAnkB. \label{eq:AndreB}
\end{align}
Thus, for all $1\le \abs{k} \le n$,
\begin{align}
S_{n,k} &= \# \DAnkB = \# \TnkB.
\end{align}
In particular,  the  two sequences $\DAnkB$ and $\TnkB$ are Arnold families
for $1\leq \abs{k} \leq n$.
\end{thm}

Hetyei\cite[Definition 8]{Het96} defined  another class of
signed Andr\'e permutations.
\begin{defn}[Hetyei]
A   \emph{signed Andr\'e permutation} is  a pair $(\varepsilon, \pi)$,
where $\pi$ is an Andr\'e permutation such that  $\varepsilon(i) = 1$

if  $\pi_i = \min \set{\pi_i, \pi_{i+1}, \dots, \pi_{n}}$.
\end{defn}
We write  $\DAnH$ (resp. $\DAnkH$) for the set of the signed Andr\'e permutations (resp. ending with entry $k$) in $\mathcal{B}_n$.
Some examples of $\DAH_{4,k}$ are shown in Table~\ref{table:DARSB}.

We have the following conjecture.
\begin{conj}

For all $1 \le k \le n$, we have
\begin{align*}
S_{n,k} = \# \DAH_{n+1,n+2-k}.
\end{align*}
\label{conj:main}
\end{conj}
Since the last entry of any permutation in the family $\DAnH$ is always positive,
even if Conjecture~\ref{conj:main} is true, it covers only the half of Table \ref{tab:Arnold}.
Now we define signed Simsun permutations corresponding to Heytei's signed Andr\'e permutations.
\begin{defn}
A permutation $\pi$ in $\mathcal{B}_n$ is a \emph{signed Simsun permutation}
if $\abs{\pi_1} \abs{\pi_2} \dots \abs{\pi_{n}}$ is a Simsun permutation
and $\pi_{i}>0$
for all $\abs{\pi_i} = \min \set{\abs{\pi_i}, \abs{\pi_{i+1}}, \dots, \abs{\pi_{n}}}$.

\end{defn}
Let $\RSnB$ be the set of signed Simsun permutations in $\mathcal{B}_n$
and $\RSnkB$ the set of signed Simsun permutations in $\RSnB$ with last entry $k$.

Some examples of $\RSB_{3,k}$ are shown in Table~\ref{table:DARSB}.

\begin{thm}
\label{thm:main3}
For positive integer $n\ge 1$, there is a bijection $\varphi^{(B)}: \DAnH \to \RSnoneB$
such that
\begin{align}
\Last(\sigma)-1=\Last(\varphi^{(B)}(\sigma))  \label{eq:SimsunB}
\end{align}
for all $\sigma \in \DAnH$.
In words, the mapping $\varphi^{(B)}$ is a bijection from $\DAnkH$ onto $\RSnkoneB$.
\end{thm}

\begin{table}[t]
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c||c|c|c|c}
$k$
& $\DUB_{3,k}$
& $\DAB_{3,k}$
& $\DAH_{4,5-k}$
& $\RSB_{3,4-k}$
\\ \hline
$1$
& $\set{1\bar{2}3, 1\bar{3}2, 1\bar{3}\bar{2}}$
& $\set{3\bar{2}1, \bar{3}\bar{2}1, 2\bar{3}1}$
& $\set{1234, 3124, \bar{3}124}$
& $\set{123, 213, \bar{2}13}$
\\
$2$
& $\set{213, 2\bar{1}3, 2\bar{3}1, 2\bar{3}\bar{1}}$
& $\set{312, \bar{3}12, 3\bar{1}2, \bar{3}\bar{1}2}$
& $\set{1423, 1\bar{4}23, 4123, \bar{4}123}$
& $\set{132, 1\bar{3}2, 312, \bar{3}12}$
\\
$3$
& $\set{312, 3\bar{1}2, 3\bar{2}1, 3\bar{2}\bar{1}}$
& $\set{\bar{2}13, \bar{2}\bar{1}3, 123, \bar{1}23}$
& $\set{3412, \bar{3}412, 3\bar{4}12, \bar{3}\bar{4}12}$
& $\set{231, \bar{2}31, 2\bar{3}1, \bar{2}\bar{3}1}$
\\
\end{tabular}\\
\vspace{1em}
\caption{The sets $\DUB_{3,k}$, $\DAB_{3,k}$, $\DAH_{4,5-k}$, and $\RSB_{3, 4-k}$ for $1\le k \le 3$}
\label{table:DARSB}
\end{table}

\begin{rmk}
Ehrenborg and Readdy \cite[Section 7]{ER98} gave a different definition of
\emph{signed Simsun permutation} as follows:
A signed permutation $\sigma$ of length $n$ is a Simsun permutation if $\sigma_{[k]}$ have no double descents for all $1\le k\le n$, where $\sigma_{[k]}$ is obtained by removing the $(n-k)$ entries $\pm (k+1), \dots, \pm n$ from $\sigma$.
Beacuse all eight signed permutations of length $2$
$$
12,~21,
~\bar{1}2, ~ 2\bar{1},
~1\bar{2}, ~ \bar{2}1,
~\bar{1}\bar{2}, ~ \bar{2}\bar{1}
$$
are Simsun permutations,
we note that it is not an Arnold family.
\end{rmk}

\section{Proof of Theorems~\ref{main2}, \ref{main1}, and \ref{thm:main3}}
\label{sec:proof1}
First of all, we prove Theorem~\ref{main2},
in order to show that $(\DAnk)_{1\le k \le n}$ is a Entringer family,
that is, $E_{n,k} = \# \DAnk$.
We construct a bijection $\omega$ between $\Tnk$ and $\DAnk$ in Section~\ref{sec:omega}.

Hence the map $\omega \circ \psi$ is a bijection from the set $\DUnk$ of (down-up) alternating permutations with first entry $k$ to the set $\DAnk$ of Andr\'e permutations with last entry $k$.
Also, in order to show that $(\RSnkone)_{1\le k \le n}$ is a Entringer family, that is $E_{n,k} = \# \RSnkone$,

we construct a bijection between $\DAnk$ and $\RSnkone$ in Section~\ref{sec:varphi} and then two sets $\DAnk$ and $\RSnkone$ have the same cardinality.

\subsection{Proof of Theorem~\ref{main2}: Bijection $\omega : \Tnk \to \DAnk$.}

\label{sec:omega}

Given $T \in \Tnk$,
write down the word $\sigma$ of the vertices of the tree $T$ in \emph{inorder}, namely,
for any vertex $v$ in $T$, the left child of $v$ and its descendants precede the vertex $v$ and the vertex $v$ precede the right child of $v$ and its descendants.
Since $T$ is an increasing tree, we can recover $T$ from $\sigma$ by finding minimum in subwords in $\sigma$ successively.
The word $\sigma$ has no double ascents because no vertex in $T$ has only a right child.
The leaf $p(T)$ of the minimal path of $T$ is $\sigma_1$ and the parent of $\sigma_1$ is $\sigma_2$, so $\sigma$ starts with a decent, that is, $\sigma_1 > \sigma_2$.
Similarly, since the subgraph of $T$ consisting of $1, \dots, k$, for any $k$, is also well-defined an increasing 1-2 subtree, the subwords of $\sigma$ consisting of $1, \dots, k$ has also no double ascents and starts with a decent.
Thus the word $\sigma^R$, which is the reverse word of $\sigma$, is an Andr\'e permutation of $n$ and let $\omega(T) = \sigma^R$.

For example, if the tree $T$ is given by the following figure with a corresponding word $\sigma$,
$$

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$$
then $\omega(T) = \sigma^R = 684512937$ as reading reversely vertices of $T$ in inorder.

\subsection{Proof of Theorem~\ref{main1}: Bijection $\varphi : \DAnk \to \RS_{n-1,k-1}$.}

\label{sec:varphi}

Given $\sigma:=\sigma_1\ldots \sigma_n \in \DAnk$, let
$i_1<\ldots< i_\ell$ be the positions of the right-to-left minima of $\sigma$.
Clearly
$\sigma_{i_1}=1$,  $i_{\ell-1}=n-1$ and  $i_\ell=n$ with $\sigma_{n}=k$.
Let $\varphi(\sigma)=\pi$, where $\pi=\pi_1 \pi_2 \dots \pi_{n-1}$ is defined by
\begin{align}
\pi_i=\begin{cases}
\sigma_i-1& \text{if}\quad i\notin\{i_1, \ldots, i_\ell\}, \\
\sigma_{i_k}-1& \text{if}\quad  i=i_{k-1} \quad \text{for}\; k=2, \ldots, \ell.
\end{cases}
\label{eq:move}
\end{align}

We show that $\pi \in \RS_{n-1,k-1}$.
Suppose $\pi_{[i]}$ has a double descent for some $1\le i \le n$.
There exists a triple $(a,b,c)$ such that
$1\le a < b < c \le n$, $i \ge \pi_a > \pi_b > \pi_c$, and
$\pi_{a+1}, \ldots, \pi_{b-1}$, $\pi_{b+1}, \ldots, \pi_{c-1}$ are greater than $i$,
which yields $\pi_c = \min\set{\pi_j : a\le j \le c}$.
Then $\pi_{c}$ could be a right-to-left minimum in $\pi$ and the others $\pi_a, \pi_{a+1}, \dots, \pi_{c-1}$ are not in $\pi$. As $\varphi(\sigma) = \pi$, we have
$$
\sigma_a = \pi_a+1,~ \sigma_{a+1} = \pi_{a+1}+1,~ \ldots, \sigma_{c-1} = \pi_{c-1}+1
\text{, and }
\sigma_{c} \le \pi_{c}+1.
$$
Hence a triple $(\sigma_{a}, \sigma_{b}, \sigma_{c})$ is a double descent in $\sigma_{[i]}$,
which contradicts that $\sigma$ is an Andr\'e permutation.
As this procedure is clearly reversible, the mapping $\varphi$ is a bijection.

Consider the running example $\sigma=684512937$. The right-to-left minimums of $\sigma$ are $1$, $2$, $3$, $7$.
So after removing $1$ from $\sigma$,
the entries $2$, $3$, $7$ are moved to the positions of $1$, $2$, $3$, respectively,
and we get the permutation $\hat\pi = 68452397$,
and then $\varphi(\sigma) = \pi = 57341286$,
which is a Simsun permutation of length $8$ with last entry $6$.

\begin{rmk}
Considering the bijection $\psi$ in Theorem~\ref{thm:GSZ11},
the map $\varphi \circ \omega \circ \psi$ is a bijection from the set of (down-up) alternating permutations of length $n$ with first entry $k$ to the set of Simsun permutations of length $n-1$ with last entry $k-1$. Namely we have the diagram in Figure~\ref{fig:diagram1}.

For example,
if $\tau = 739154826 \in \DU_{9,7}$, then
\begin{align*}
\psi(\tau) &= T  \in T_{9,7}, \\
\omega(T) &= \sigma = 684512937 \in \DA_{9,7}, \\
\varphi(\sigma) &= \pi = 57341286 \in \RS_{8,6},
\end{align*}
where $T$ is the increasing 1-2 tree given in Section~\ref{sec:omega}.

\end{rmk}

\subsection{Proof of Theorem~\ref{thm:main3}}
One can extend the above mapping $\varphi$ defined on $\DAn$ to a mapping $\varphi^{(B)}$ on $\DAnH$.
It is also bijective between $\DAnkH$ and $\RSnkoneB$,
but the description of $\varphi^{(B)}$ and a proof of bijectivity are omitted,
because it is very similar in Section~\ref{sec:varphi}.

\begin{rmk}
This bijection preserves the $cd$-indices between Andr\'e permutations and Simsun permutations.
The \emph{variation} of a permutation $\pi=\pi_1 \dots \pi_n$ is given by $\a\b$-monomial $u_1\dots u_{n-1}$
such that $u_i=\a$ if $\pi_i < \pi_{i+1}$ and $u_i=\b$ if $\pi_i > \pi_{i+1}$.
The \emph{reduced variation} of Andr\'e permuation is defined by replacing each $\b\a$ with $\d$ and then replacing each remaining $\a$ by $\c$.
For example, the variation and reduced variation of Andr\'e permutation $\sigma=684512937$ is
$$\texttt{ababaaba} = \texttt{cddcd}.$$

For the cd-index of a Simsun permutation $\sigma$,
we consider \emph{augmented} Simsun permutation by adding $\sigma(0)=0$.
Here, the \emph{reduced variation} of augmented Simsun permuations is defined by replacing each $\a\b$ with $\d$ and then replacing each remaining $\a$ by $\c$.
So the variation and reduced variation of augmented Simsun permutation $057341286$ by adding $0$ to $\varphi(\sigma) = 57341286$ is
$$\texttt{aababaab} = \texttt{cddcd}.$$

\end{rmk}

\section{Proof of Theorem~\ref{thm:main2}}
\label{sec:proof2}

Given a $\sigma \in \DUnkB$,
there is a unique order-preserving map $\pi_\sigma$, say just $\pi$, from $\set{\sigma_1, \dots, \sigma_n}$ to $[n]$.
In other words, $\pi$ replaces the $i$-th smallest entry in $\sigma$ by $i$.
The permutation $\tau = \pi \sigma$ belongs to $\Ank$ and $\psi(\tau) = \psi(\pi \sigma)$ in $\Tnk$. Then $\pi^{-1}(\psi(\tau))$ means the tree with vertex labelings $\set{\sigma_1, \dots, \sigma_n}$ instead of $[n]$ and it should belong to $\TnkB$.
So we construct the bijection $\psi^{B}$ from $\DUnkB$ to $\TnkB$ by
$$\psi^{B}(\sigma) = \pi^{-1}(\psi(\pi \sigma))$$
through the unique order-preserving map $\pi$.
Hence, it yields \eqref{eq:TreeB}.

For example, in the case of $\sigma = 6\bar{3}9\bar{8}2\bar{1}7\bar{4}5$,
the order-preserving map $\pi_\sigma$ is
$$\pi = {\bar{8}\bar{4}\bar{3}\bar{1}25679 \choose 123456789}.$$
So we have $\tau = \pi \sigma = 739154826$ and $\psi(\tau) = \psi(\pi \sigma)$ and $\psi^{B}(\sigma) = \pi^{-1}(\psi(\pi \sigma))$ are illustrated as
$$

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\pgfcircle[fill]{\pgfxy(10.00,10.00)}{0.70mm}
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\pgfcircle[stroke]{\pgfxy(30.00,0.00)}{0.70mm}
\pgfcircle[fill]{\pgfxy(80.00,10.00)}{0.70mm}
\pgfcircle[stroke]{\pgfxy(80.00,10.00)}{0.70mm}
\pgfcircle[fill]{\pgfxy(70.00,20.00)}{0.70mm}
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\pgfcircle[fill]{\pgfxy(60.00,0.00)}{0.70mm}
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\pgfputat{\pgfxy(-20.00,0.00)}{\pgfbox[bottom,left]{\fontsize{9.96}{11.95}\selectfont $\psi(\tau)=$}}
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\pgfputat{\pgfxy(170.00,4.00)}{\pgfbox[bottom,left]{\fontsize{7.97}{9.56}\selectfont \makebox[0pt]{$2$}}}
\pgfputat{\pgfxy(180.00,-6.00)}{\pgfbox[bottom,left]{\fontsize{7.97}{9.56}\selectfont \makebox[0pt]{$\bar{1}$}}}
\pgfputat{\pgfxy(130.00,4.00)}{\pgfbox[bottom,left]{\fontsize{7.97}{9.56}\selectfont \makebox[0pt]{$\bar{3}$}}}
\pgfputat{\pgfxy(150.00,-6.00)}{\pgfbox[bottom,left]{\fontsize{7.97}{9.56}\selectfont \makebox[0pt]{$\bar{4}$}}}
\pgfputat{\pgfxy(160.00,-16.00)}{\pgfbox[bottom,left]{\fontsize{7.97}{9.56}\selectfont \makebox[0pt]{$\bar{8}$}}}
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\pgfcircle[fill]{\pgfxy(130.00,10.00)}{0.70mm}
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\pgfcircle[fill]{\pgfxy(170.00,10.00)}{0.70mm}
\pgfcircle[stroke]{\pgfxy(170.00,10.00)}{0.70mm}
\pgfcircle[fill]{\pgfxy(150.00,0.00)}{0.70mm}
\pgfcircle[stroke]{\pgfxy(150.00,0.00)}{0.70mm}
\pgfcircle[fill]{\pgfxy(200.00,10.00)}{0.70mm}
\pgfcircle[stroke]{\pgfxy(200.00,10.00)}{0.70mm}
\pgfcircle[fill]{\pgfxy(190.00,20.00)}{0.70mm}
\pgfcircle[stroke]{\pgfxy(190.00,20.00)}{0.70mm}
\pgfcircle[fill]{\pgfxy(180.00,0.00)}{0.70mm}
\pgfcircle[stroke]{\pgfxy(180.00,0.00)}{0.70mm}
\pgfputat{\pgfxy(100.00,0.00)}{\pgfbox[bottom,left]{\fontsize{9.96}{11.95}\selectfont $\psi^B(\sigma)=$}}
\pgfcircle[fill]{\pgfxy(160.00,-10.00)}{0.70mm}
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\pgfputat{\pgfxy(190.00,14.00)}{\pgfbox[bottom,left]{\fontsize{7.97}{9.56}\selectfont \makebox[0pt]{$7$}}}
\pgfputat{\pgfxy(84.00,0.00)}{\pgfbox[bottom,left]{\fontsize{9.96}{11.95}\selectfont ,}}
\end{pgfpicture}

$$

In Subsection \ref{sec:omega}, we define the bijection $\omega$ from $\Tnk$ to $\DAnk$. Given a tree $T$ in $\TnkB$, there is a unique order-preserving map $\pi_T$, say just $\pi$, from $V(T)$ to $[n]$.
In other words, $\pi$ replaces the $i$-th smallest $V(T)$ by $i$.
After relabeling on vertices of $T$ by $\pi$, we obtain the tree $\pi(T)$ which belongs to $\Tnk$ and $\omega(\pi(T))$ is in $\DAnk$. Then $\pi^{-1}(\omega(\pi(T)))$ should belong to $\DAnkB$.
So we construct the bijection $\omega^{B}$ from $\TnkB$ to $\DAnkB$ by
$$\omega^{B}(T) = \pi^{-1}(\omega(\pi(T)))$$
through the unique order-preserving map $\pi$.
Such the map $\omega^{B}$ can be described simply, as same as $\omega$, by reading reversely vertices of $T$ in inorder.
Hence, it yields \eqref{eq:AndreB}.

For example, in the case of $T$ illustrated as
$$

\centering
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\pgfputat{\pgfxy(80.00,-26.00)}{\pgfbox[bottom,left]{\fontsize{7.97}{9.56}\selectfont \makebox[0pt]{$5$}}}
\pgfputat{\pgfxy(70.00,-26.00)}{\pgfbox[bottom,left]{\fontsize{7.97}{9.56}\selectfont \makebox[0pt]{$7$}}}
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\end{pgfpicture}

$$
we obtain $\omega^B(T) = \sigma^R = 57\bar{1}2\bar{8}\bar{4}9\bar{3}6$ by reading reversely vertices of $T$ in inorder.
So bijections for type A and type B commute in the diagram of Figure~\ref{fig:diagram1}.
\begin{figure}[t]
\[
\xymatrix{
\DUnk    \ar[r]^{\psi}
&\Tnk    \ar[r]^{\omega}
&\DAnk  \ar[rr]^-{\varphi}
&&\RSnkone \\
\DUnkB   \ar[u]^-{\pi} \ar[r]^-{\psi^{B}}
&\TnkB   \ar[u]^-{\pi} \ar[r]^-{\omega^{B}}
& \DAnkB \ar[u]^-{\pi} \ar@{.}[r]^-{?}
& \DAH_{n+1, n+2-k} \ar[r]^-{\varphi^{(B)}}
& \RSB_{n, n+1-k}
}
\]
\caption{Bijections between Entringer families and Arnold families}
\label{fig:diagram1}
\end{figure}

We summarize four interpretations for Entringer numbers $E_{4,k}$, $k \in \set{2,3,4}$
in Table~\ref{tab:A} and
left three interpretations for Arnold number $S_{3,k}$, $k \in \set{1,2,3}$ in Table~\ref{tab:B}.
In every column, the corresponding elements are described via the different bijections mentioned in the paper.

\begin{table}[t]
\centering
\begin{tabular}{c||c|>{\centering}m{1in}|c|c}
$k$ & $\tau \in \DU_{4,k}$ & $\psi(\tau) \in \T_{4,k}$ & $\omega(\psi(\tau)) \in \DA_{4,k}$ & $\varphi(\omega(\psi(\tau))) \in \RS_{3,k-1}$ \\ \hline
\multirow{1}{*}{$2$} & $2143$ &

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												  	& $3412$ & $231$ \\ \hline
\multirow{2}{*}{$3$} & $3241$ &

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\pgfmoveto{\pgfxy(55.00,10.00)}\pgflineto{\pgfxy(60.00,5.00)}\pgfstroke
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\pgfcircle[fill]{\pgfxy(55.00,10.00)}{0.42mm}
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\pgfcircle[fill]{\pgfxy(65.00,10.00)}{0.42mm}
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\pgfmoveto{\pgfxy(65.00,10.00)}\pgflineto{\pgfxy(60.00,5.00)}\pgfstroke
\pgfputat{\pgfxy(67.00,9.00)}{\pgfbox[bottom,left]{\fontsize{7.97}{9.56}\selectfont 4}}
\end{pgfpicture}

												 	& $1423$ & $132$ \\ \cline{2-5}
                     & $3142$ &

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\pgfmoveto{\pgfxy(25.00,10.00)}\pgflineto{\pgfxy(30.00,5.00)}\pgfstroke
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                     							  	& $4123$ & $312$ \\ \hline
\multirow{2}{*}{$4$} & $4231$ &

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													& $1234$ & $123$ \\ \cline{2-5}
                     & $4132$ &

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                     								& $3124$ & $213$ \\
\end{tabular}
\vspace{1em}
\caption{Three bijections between Entringer families with $n=4$ and $2\le k \le 4$}
\label{tab:A}
\end{table}

\begin{table}[t]
\centering
\renewcommand{\arraystretch}{0}
\begin{tabular}{c|>{\centering}m{1.2in}|c||c||c|c}
$\tau \in \DUB_{3,k}$ & $\psi^{(B)}(\tau) \in \TB_{3,k}$ & $\omega^{(B)}(\psi^{(B)}(\tau)) \in \DAB_{3,k}$
& $k$ & $\sigma \in \DAH_{4,5-k}$ & $\varphi^{(B)}(\sigma) \in \RSB_{3,4-k}$ \\ \hline
$1\bar{2}3$       &

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& $3\bar{2}1$       & \multirow{3}{*}{$1$} & $1234$             & $123$             \\ \cline{1-3} \cline{5-6}
$1\bar{3}2$       &

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& $2\bar{3}1$       &                      & $3124$             & $213$             \\ \cline{1-3} \cline{5-6}
$1\bar{3}\bar{2}$ &

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& $\bar{3}\bar{2}1$ &                      & $\bar{3}124$       & $\bar{2}13$       \\ \hline
$213$             &

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& $312$             & \multirow{4}{*}{$2$} & $1423$             & $132$             \\ \cline{1-3} \cline{5-6}
$2\bar{1}3$       &

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& $3\bar{1}2$       &                      & $1\bar{4}23$       & $1\bar{3}2$       \\ \cline{1-3} \cline{5-6}
$2\bar{3}1$       &

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& $\bar{3}12$       &                      & $4123$             & $312$             \\ \cline{1-3} \cline{5-6}
$2\bar{3}\bar{1}$ &

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& $\bar{3}\bar{1}2$ &                      & $\bar{4}123$       & $\bar{3}12$       \\ \hline
$312$             &

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& $123$             & \multirow{4}{*}{$3$} & $3412$             & $231$             \\ \cline{1-3} \cline{5-6}
$3\bar{1}2$       &

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& $\bar{1}23$       &                      & $\bar{3}412$       & $\bar{2}31$       \\ \cline{1-3} \cline{5-6}
$3\bar{2}1$       &

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& $\bar{2}13$       &                      & $3\bar{4}12$       & $2\bar{3}1$       \\ \cline{1-3} \cline{5-6}
$3\bar{2}\bar{1}$ &

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& $\bar{2}\bar{1}3$ &                      & $\bar{3}\bar{4}12$ & $\bar{2}\bar{3}1$ \\ \hline
\end{tabular}
\vspace{1em}
\caption{Three bijections between Arnold families with $n=3$ and $1\le k \le 3$}
\label{tab:B}
\end{table}

\section{New description of two known bijections}
\label{sec:psi}

\subsection{A decomposition  of  Chuang et al.'s bijection $\phi: \T_{n+1} \to \RS_{n}$}
In 2012, Chuang et al.~\cite{CEFP12} construct a bijection $\phi: \T_{n+1} \to \RS_{n}$.
If $x$ has only one child then it is the right child of $x$.
They described the bijection $\phi$ between increasing 1-2 trees and Simsun permutations using
the following algorithm.

\subsection*{Algorithm~A}
\begin{itemize}
\item[(A1)] If $T$ consists of the root vertex then $T$ is associated with an empty word.
\item[(A2)] Otherwise, the word $\rho(T)$ is defined inductively by the factorization
$$
\rho(T)=\omega\cdot \rho(T'),
$$
where the subword $\omega$ and the subtree $T'$ are determined as follows.
\begin{enumerate}[(a)]
\item
If the root of $T$ has only one child $x$ then let $\omega = x$ (consisting of a single
letter $x$), let $T'=\tau(x)$ (i.e, obtained from $T$ by deleting the root of $T$), and
relabel the vertex $x$ by $1$.
\item
If the root of $T$ has two children $u$, $v$ with $u>v$
then traverse the right subtree $\tau(u)$ reversely in inorder,
put down the word $\omega$ of the vertices of $\tau(u)$ and let $T'=T-\tau(u)$.
\end{enumerate}
\end{itemize}
As deleted only $1$ in $\rho(T)$,
every permutation $\rho(T)=a_1 a_2\cdots a_n$ is a Simsun permutation on $\set{2, 3, \dots, n+1}$.
Thus we get a Simsun permutation $\phi(T)=b_1 b_2\cdots b_n$ on $[n]$ with $b_i = a_i - 1$ for all $1\le i \le n$.

\begin{rmk}
Originally, in \cite{CEFP12}, the increasing 1-2 trees on $n$ vertices are labeled with $0,1, \ldots, n-1$ instead of $1,2, \ldots, n$ and was drawn in a \emph{canonical form} such that if a vertex $x$ has two children $u$, $v$ with $u > v$ then $u$ is the left child, $v$ is the right child.

\end{rmk}

\begin{thm}
The bijection $\phi : \Tnk \to \RSnkone$ is the composition of $\varphi$ and $\omega$; i.e.,  $\phi = \varphi \circ \omega$.
\end{thm}

\begin{proof}
Suppose that we let $\omega$ be the root of $T$ without relabeling the vertex $x$ in (A2a),
it is obvious that Algorithm~A follow the bijeciton $\omega$, i.e.,
reading the vertex of $T$ reversely in inorder.
So it is enough to show that the change in two rules of (A2a) follows $\varphi$.

The root of $T'$ becomes to the left-child of the root of $T$ after (A2a) and
the root of $T'$ is same with the root of $T$ in (A2b).
So Algorihm A executes the step (A2a)
only when a vertex on the minimal path of an original tree becomes the root of its subtree.

To record the left-child $x$ instead of the root $1$ with relabeling the vertex $x$ by $1$ in each of (A2a)'s means
to exchange $x$ and $1$ sequentially according to the minimal path from the root $1$ of the original tree $T$.

It is clear that all vertices in the minimal path in a tree become the right-to-left minimums in a permutation under $\omega$. So each right-to-left minimums is recorded in the position of the previous right-to-left minimums in a permutation obtained from a tree by $\omega$.
Since all elements are decreased by 1 in the last step,
it satisfies \eqref{eq:move},
then Algorithm~A follows $\varphi \circ \omega$.
\end{proof}

\subsection{A new description of Gelineau et al.'s
bijection $\psi: \Ank\to \Tnk$}
The  bijection $\psi$ between $\Ank$ and $\Tnk$  was
constructed  as a composition of two bijections via the set $\mathcal{ES}_{n,k}$ of \emph{encoding sequences}  in \cite{GSZ11}.
In this section, we just give directly another description of this bijection $\psi$ from $\Ank$ to $\Tnk$, which does not use encoding sequences.

Given an increasing 1-2 tree $T \in T_n$, by convention,
if a vertex $x$ of $T$ has two children $u$, $v$ with $u<v$ then $u$ is the \emph{left child} and $v$ is the \emph{right child}.
By convention, if $x$ has only one child then it is the left child of $x$.

\subsection*{Algorithm~B}
Gelineau et al. described the bijection $\psi$ between alternating permutations and increasing 1-2 trees using the following algorithm.

Due to, for $n=1$ or $2$,
$$\abs{\A_n}=\abs{\T_n}=1,$$
we can define trivially $\psi$.
For $n\ge 3$, given $\pi \in \Ank$ ($k=\pi_1$), we define the mapping $\psi:\Ank \to \Tnk$ recursively as follows:
\begin{enumerate}[(B1)]
\item \label{case:a} If $\pi_2=k-1$, then define
$\pi' = \pi'_1\pi'_2 \dots \pi'_{n-2} \in \A_{n-2,i-2}$
by deleting $k-1$ and $k$ from $\pi$
and relabeling by $[n-2]$ where $i> k$, that is, for all $1\le j \le n-2$
$$\pi'_{j} =
\begin{cases}
\pi_{j+2},   &\text{if $\pi_{j+2}<k-1$,}\\
\pi_{j+2}-2, &\text{if $\pi_{j+2}>k$.}
\end{cases}
$$
We get $T'=\psi(\pi') \in \T_{n-2,i-2}$.
Relabel $T'$ by $\set{1,\dots,k-2,k+1,\dots,n}$ keeping in the order of labels,
denoted by $T''$. Let $m$ be the smallest vertex greater than $k$ in the minimal path of $T''$ and $\ell$ the parent of $m$ in $T''$. Then insert a vertex $k-1$ in the middle of the edge $(\ell,m)$ and graft $k$ as the left-child of $k-1$.
$$

\centering
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$$
We get the tree $T = \psi(\pi) \in \Tnk$.

\item \label{case:b}
If $\pi_2<k-1$, then define $\pi'=(k-1~k)\pi \in \A_{n,k-1}$
by exchanging $k-1$ and $k$ in $\pi$.
We get $T'=\psi(\pi') \in \T_{n,k-1}$.

\begin{enumerate}[(a)]
\item If $k$ is a not sibling of $k-1$ in $T'$, then we get the tree $T = \psi(\pi) \in \Tnk$ exchanging the labels $k-1$ and $k$ in $T'$. \label{case:b2}
$$

\centering
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$$

\item If $k$ is a sibling of $k-1$ in $T'$, then we get the tree $T = \psi(\pi) \in \Tnk$ modifying as follows: \label{case:b1}
$$

\centering
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$$
\end{enumerate}
\end{enumerate}

\subsection*{Algorithm~C}
We define another bijection $\psi' : \Ank \to \Tnk$ by the following algorithm.
Given $\sigma = \sigma_1 \dots \sigma_n \in \Ank$,
denote by $d_i(\sigma) = (\sigma_{2i-1}, \sigma_{2i})$ with $\sigma_{2i-1} > \sigma_{2i}$
for $1 \le i \le m$,
where $m = \lfloor (n+1)/2\rfloor$, but $d_{m}(\sigma) = (\sigma_{n})$ if $n$ is odd.
We shall construct a series of  trees $T^{(m)}, \ldots, T^{(1)}$
by reading the pairs $d_m(\sigma)$, \dots, $d_1(\sigma)$ in this order.

If $n$ is odd, so $d_{m} (\sigma) = (\sigma_n)$,
define $T^{(m)}$ to be the tree with only root $\sigma_n$.
If $n$ is even, so $d_{m}(\sigma) = (\sigma_{n-1}, \sigma_{n})$ with $\sigma_{n-1} > \sigma_{n}$,
define $T^{(m)}$ to be  the tree with the root $\sigma_{n}$ and its left child $\sigma_{n-1}$.

We note that the vertex $\sigma_{2m-1}$ is the minimal leaf of $T^{(m)}$.

For $1 \le i \le m-1$,
assuming that we have a tree $T^{(i+1)}$ of which the minimal leaf is $\sigma_{2i+1}$,
read $d_i(\sigma) = (\sigma_{2i-1}, \sigma_{2i})$.
As $\sigma_{2i} < \sigma_{2i+1}$,
the smallest vertex, say $a^{(i)}$, greater than $\sigma_{2i}$ in the minimal path of $T^{(i+1)}$ is well-defined.
By removing all left edges from the increasing 1-2 tree $T^{(i+1)}$, we have several paths, each connected component of which is called a \emph{maximal path} consisting only right edges.

\begin{enumerate}[(C1)]
\item
If $a^{(i)} < \sigma_{2i-1}$,
the largest vertex, say $b^{(i)}$, less than $\sigma_{2i-1}$ in the maximal path from $a^{(i)}$ of $T^{(i+1)}$ is well-defined.
For some $j\ge 1$, let $(v_1, v_1, \ldots, v_j)$ be the path
from $a^{(i)}$ to $b^{(i)}$ with $v_1=a^{(i)}$ and $v_j=b^{(i)}$.
The vertices $v_1, \dots, v_{j-1}$ should have left child $u_1, \dots, u_{j-1}$, but $v_j$ may not, with
\begin{align*}
v_1 < u_1 < v_2 < u_2 < \dots < v_{j-1} < u_{j-1} < v_j
\end{align*}
Decomposing by the maximal path from $a^{(i)}$ to $b^{(i)}$,
we write $S_1, \ldots, S_{j}$ the left-subtrees of the vertices $v_1, \ldots, v_{j}$
and $S_{j+1}$  the right-subtree of $b^{(i)}$.
Since each of $u_1, \dots, u_{j-1}$ lies on each of $S_1, \dots, S_{j-1}$,
$S_1, \dots, S_{j-1}$ should not be empty, but two trees $S_{j}$ and $S_{j+1}$ may be empty.
Then, the tree $T^{(i)}$ is obtained from $T^{(i+1)}$ by the following operations:
\begin{itemize}
\item Graft $\sigma_{2i}$ so that $a^{(i)}$ is its left-child;
\item Flip the tree at vertex $a^{(i)}$;
\item Transplant the trees $S_1, \ldots, S_{j+1}$ as right-subtrees of
the vertices $\sigma_{2i}$, $a^{(i)}$, $v_2, \ldots, v_{j-1}$, $b^{(i)}$;
\item Graft $\sigma_{2i-1}$ as the left-child of $b^{(i)}$.
\end{itemize}
We can illustrate the above transformation by
$$

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\pgfmoveto{\pgfxy(130.00,70.00)}\pgfcurveto{\pgfxy(133.29,72.07)}{\pgfxy(136.63,74.07)}{\pgfxy(140.00,76.00)}\pgfcurveto{\pgfxy(144.15,78.38)}{\pgfxy(148.39,80.67)}{\pgfxy(153.00,82.00)}\pgfcurveto{\pgfxy(156.32,82.96)}{\pgfxy(160.00,84.27)}{\pgfxy(160.00,87.50)}\pgfcurveto{\pgfxy(160.00,89.99)}{\pgfxy(157.62,91.65)}{\pgfxy(155.00,92.00)}\pgfcurveto{\pgfxy(151.89,92.42)}{\pgfxy(148.85,91.30)}{\pgfxy(146.00,90.00)}\pgfcurveto{\pgfxy(140.92,87.69)}{\pgfxy(136.05,84.68)}{\pgfxy(133.00,80.00)}\pgfcurveto{\pgfxy(131.06,77.02)}{\pgfxy(130.02,73.55)}{\pgfxy(130.00,70.00)}\pgfstroke
\pgfmoveto{\pgfxy(50.00,-11.00)}\pgfcurveto{\pgfxy(53.29,-8.93)}{\pgfxy(56.63,-6.93)}{\pgfxy(60.00,-5.00)}\pgfcurveto{\pgfxy(64.15,-2.62)}{\pgfxy(68.39,-0.33)}{\pgfxy(73.00,1.00)}\pgfcurveto{\pgfxy(76.32,1.96)}{\pgfxy(80.00,3.27)}{\pgfxy(80.00,6.50)}\pgfcurveto{\pgfxy(80.00,8.99)}{\pgfxy(77.62,10.65)}{\pgfxy(75.00,11.00)}\pgfcurveto{\pgfxy(71.89,11.42)}{\pgfxy(68.85,10.30)}{\pgfxy(66.00,9.00)}\pgfcurveto{\pgfxy(60.92,6.69)}{\pgfxy(56.05,3.68)}{\pgfxy(53.00,-1.00)}\pgfcurveto{\pgfxy(51.06,-3.98)}{\pgfxy(50.02,-7.45)}{\pgfxy(50.00,-11.00)}\pgfstroke
\pgfmoveto{\pgfxy(30.00,10.00)}\pgfcurveto{\pgfxy(33.29,12.07)}{\pgfxy(36.63,14.07)}{\pgfxy(40.00,16.00)}\pgfcurveto{\pgfxy(44.15,18.38)}{\pgfxy(48.39,20.67)}{\pgfxy(53.00,22.00)}\pgfcurveto{\pgfxy(56.32,22.96)}{\pgfxy(60.00,24.27)}{\pgfxy(60.00,27.50)}\pgfcurveto{\pgfxy(60.00,29.99)}{\pgfxy(57.62,31.65)}{\pgfxy(55.00,32.00)}\pgfcurveto{\pgfxy(51.89,32.42)}{\pgfxy(48.85,31.30)}{\pgfxy(46.00,30.00)}\pgfcurveto{\pgfxy(40.92,27.69)}{\pgfxy(36.05,24.68)}{\pgfxy(33.00,20.00)}\pgfcurveto{\pgfxy(31.06,17.02)}{\pgfxy(30.02,13.55)}{\pgfxy(30.00,10.00)}\pgfstroke
\pgfmoveto{\pgfxy(60.00,60.00)}\pgfcurveto{\pgfxy(63.29,62.07)}{\pgfxy(66.63,64.07)}{\pgfxy(70.00,66.00)}\pgfcurveto{\pgfxy(74.15,68.38)}{\pgfxy(78.39,70.67)}{\pgfxy(83.00,72.00)}\pgfcurveto{\pgfxy(86.32,72.96)}{\pgfxy(90.00,74.27)}{\pgfxy(90.00,77.50)}\pgfcurveto{\pgfxy(90.00,79.99)}{\pgfxy(87.62,81.65)}{\pgfxy(85.00,82.00)}\pgfcurveto{\pgfxy(81.89,82.42)}{\pgfxy(78.85,81.30)}{\pgfxy(76.00,80.00)}\pgfcurveto{\pgfxy(70.92,77.69)}{\pgfxy(66.05,74.68)}{\pgfxy(63.00,70.00)}\pgfcurveto{\pgfxy(61.06,67.02)}{\pgfxy(60.02,63.55)}{\pgfxy(60.00,60.00)}\pgfstroke
\end{pgfpicture}

$$
\item If $\sigma_{2i-1}<a^{(i)}$, then $b^{(i)}$ does not exist.
Let $S$ be  the subtree with root $a^{(i)}$ of $T^{(i+1)}$, then
the tree $T^{(i)}$ is defined as follows.
\begin{itemize}
\item Graft $\sigma_{2i}$ so that  $a^{(i)}$ is its right-child;
\item Transplant the trees $S$ as the right-subtree of the vertex $\sigma_{2i}$;
\item Graft $\sigma_{2i-1}$ as the left-child of $\sigma_{2i}$.
\end{itemize}
We can illustrate this transformation by the following
$$

\centering
\begin{pgfpicture}{8.75mm}{-13.48mm}{112.00mm}{34.41mm}
\pgfsetxvec{\pgfpoint{0.55mm}{0mm}}
\pgfsetyvec{\pgfpoint{0mm}{0.55mm}}
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\pgfputat{\pgfxy(25.00,55.00)}{\pgfbox[bottom,left]{\fontsize{7.82}{9.39}\selectfont \makebox[0pt]{$T^{(i+1)}$}}}
\pgfcircle[fill]{\pgfxy(140.00,20.00)}{0.55mm}
\pgfcircle[stroke]{\pgfxy(140.00,20.00)}{0.55mm}
\pgfmoveto{\pgfxy(140.00,20.00)}\pgflineto{\pgfxy(150.00,10.00)}\pgfstroke
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\pgfmoveto{\pgfxy(140.00,20.00)}\pgflineto{\pgfxy(130.00,30.00)}\pgfstroke
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\pgfputat{\pgfxy(137.00,17.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt][r]{$\sigma_{2i}$}}}
\pgfputat{\pgfxy(127.00,27.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt][r]{$\sigma_{2i-1}$}}}
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\pgfputat{\pgfxy(30.00,34.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt]{$S$}}}
\pgfmoveto{\pgfxy(30.00,20.00)}\pgfcurveto{\pgfxy(25.27,25.87)}{\pgfxy(21.86,32.69)}{\pgfxy(20.00,40.00)}\pgfcurveto{\pgfxy(19.57,41.67)}{\pgfxy(19.25,43.45)}{\pgfxy(20.00,45.00)}\pgfcurveto{\pgfxy(21.64,48.39)}{\pgfxy(26.11,48.00)}{\pgfxy(30.00,48.00)}\pgfcurveto{\pgfxy(33.89,48.00)}{\pgfxy(38.36,48.39)}{\pgfxy(40.00,45.00)}\pgfcurveto{\pgfxy(40.75,43.45)}{\pgfxy(40.43,41.67)}{\pgfxy(40.00,40.00)}\pgfcurveto{\pgfxy(38.14,32.69)}{\pgfxy(34.73,25.87)}{\pgfxy(30.00,20.00)}\pgfstroke
\pgfsetlinewidth{1.20mm}\pgfmoveto{\pgfxy(90.00,20.00)}\pgflineto{\pgfxy(110.00,20.00)}\pgfstroke
\pgfmoveto{\pgfxy(110.00,20.00)}\pgflineto{\pgfxy(107.20,20.70)}\pgflineto{\pgfxy(107.20,19.30)}\pgflineto{\pgfxy(110.00,20.00)}\pgfclosepath\pgffill
\pgfmoveto{\pgfxy(110.00,20.00)}\pgflineto{\pgfxy(107.20,20.70)}\pgflineto{\pgfxy(107.20,19.30)}\pgflineto{\pgfxy(110.00,20.00)}\pgfclosepath\pgfstroke
\pgfsetlinewidth{0.30mm}\pgfmoveto{\pgfxy(40.00,10.00)}\pgflineto{\pgfxy(60.00,-10.00)}\pgfstroke
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\pgfcircle[stroke]{\pgfxy(60.00,-10.00)}{0.55mm}
\pgfputat{\pgfxy(60.00,-20.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt]{$\min(T^{(i+1)})$}}}
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\pgfsetdash{}{0mm}\pgfsetlinewidth{0.30mm}\pgfmoveto{\pgfxy(60.00,-11.00)}\pgfcurveto{\pgfxy(63.29,-8.93)}{\pgfxy(66.63,-6.93)}{\pgfxy(70.00,-5.00)}\pgfcurveto{\pgfxy(74.15,-2.62)}{\pgfxy(78.39,-0.33)}{\pgfxy(83.00,1.00)}\pgfcurveto{\pgfxy(86.32,1.96)}{\pgfxy(90.00,3.27)}{\pgfxy(90.00,6.50)}\pgfcurveto{\pgfxy(90.00,8.99)}{\pgfxy(87.62,10.65)}{\pgfxy(85.00,11.00)}\pgfcurveto{\pgfxy(81.89,11.42)}{\pgfxy(78.85,10.30)}{\pgfxy(76.00,9.00)}\pgfcurveto{\pgfxy(70.92,6.69)}{\pgfxy(66.05,3.68)}{\pgfxy(63.00,-1.00)}\pgfcurveto{\pgfxy(61.06,-3.98)}{\pgfxy(60.02,-7.45)}{\pgfxy(60.00,-11.00)}\pgfstroke
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\pgfcircle[fill]{\pgfxy(150.00,10.00)}{0.55mm}
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\pgfcircle[stroke]{\pgfxy(170.00,-10.00)}{0.55mm}
\pgfputat{\pgfxy(170.00,-19.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt]{$\min(T^{(i)})$}}}
\pgfsetdash{{0.60mm}{0.50mm}}{0mm}\pgfsetlinewidth{0.60mm}\pgfmoveto{\pgfxy(171.00,-1.00)}\pgflineto{\pgfxy(161.00,9.00)}\pgfstroke
\pgfsetdash{}{0mm}\pgfsetlinewidth{0.30mm}\pgfmoveto{\pgfxy(150.00,10.00)}\pgflineto{\pgfxy(170.00,-10.00)}\pgfstroke
\pgfmoveto{\pgfxy(170.00,-11.00)}\pgfcurveto{\pgfxy(173.29,-8.93)}{\pgfxy(176.63,-6.93)}{\pgfxy(180.00,-5.00)}\pgfcurveto{\pgfxy(184.15,-2.62)}{\pgfxy(188.39,-0.33)}{\pgfxy(193.00,1.00)}\pgfcurveto{\pgfxy(196.32,1.96)}{\pgfxy(200.00,3.27)}{\pgfxy(200.00,6.50)}\pgfcurveto{\pgfxy(200.00,8.99)}{\pgfxy(197.62,10.65)}{\pgfxy(195.00,11.00)}\pgfcurveto{\pgfxy(191.89,11.42)}{\pgfxy(188.85,10.30)}{\pgfxy(186.00,9.00)}\pgfcurveto{\pgfxy(180.92,6.69)}{\pgfxy(176.05,3.68)}{\pgfxy(173.00,-1.00)}\pgfcurveto{\pgfxy(171.06,-3.98)}{\pgfxy(170.02,-7.45)}{\pgfxy(170.00,-11.00)}\pgfstroke
\pgfmoveto{\pgfxy(150.00,10.00)}\pgfcurveto{\pgfxy(153.29,12.07)}{\pgfxy(156.63,14.07)}{\pgfxy(160.00,16.00)}\pgfcurveto{\pgfxy(164.15,18.38)}{\pgfxy(168.39,20.67)}{\pgfxy(173.00,22.00)}\pgfcurveto{\pgfxy(176.32,22.96)}{\pgfxy(180.00,24.27)}{\pgfxy(180.00,27.50)}\pgfcurveto{\pgfxy(180.00,29.99)}{\pgfxy(177.62,31.65)}{\pgfxy(175.00,32.00)}\pgfcurveto{\pgfxy(171.89,32.42)}{\pgfxy(168.85,31.30)}{\pgfxy(166.00,30.00)}\pgfcurveto{\pgfxy(160.92,27.69)}{\pgfxy(156.05,24.68)}{\pgfxy(153.00,20.00)}\pgfcurveto{\pgfxy(151.06,17.02)}{\pgfxy(150.02,13.55)}{\pgfxy(150.00,10.00)}\pgfstroke
\end{pgfpicture}

$$
\end{enumerate}
We note that the vertex $\sigma_{2i-1}$ is the minimal leaf of $T^{(i)}$ for $1 \le i \le m-1$.
And then, we define $\psi'(\sigma) = T^{(1)}$.
\begin{ex}
We run the new algorithm to the examples $\sigma = 748591623$ in Example 3.2 and Fig. 2 in \cite{GSZ11}.
As $n=9$, we have five pairs
\begin{align*}
d_5(\sigma) &= (3),&
d_4(\sigma) &= (6,2),&
d_3(\sigma) &= (9,1),&
d_2(\sigma) &= (8,5),&
d_1(\sigma) &= (7,4).
\end{align*}
By Algorithm~C, we get five trees sequentially
\begin{align*}

\centering
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\pgfsetxvec{\pgfpoint{0.80mm}{0mm}}
\pgfsetyvec{\pgfpoint{0mm}{0.80mm}}
\color[rgb]{0,0,0}\pgfsetlinewidth{0.30mm}\pgfsetdash{}{0mm}
\pgfcircle[fill]{\pgfxy(0.00,0.00)}{0.48mm}
\pgfcircle[stroke]{\pgfxy(0.00,0.00)}{0.48mm}
\pgfputat{\pgfxy(-2.00,-1.00)}{\pgfbox[bottom,left]{\fontsize{9.10}{10.93}\selectfont \makebox[0pt][r]{3}}}
\pgfputat{\pgfxy(-2.50,-10.00)}{\pgfbox[bottom,left]{\fontsize{11.38}{13.66}\selectfont \makebox[0pt]{$T^{(5)}$}}}
\end{pgfpicture}

&&

\centering
\begin{pgfpicture}{-13.20mm}{-10.86mm}{2.48mm}{11.71mm}
\pgfsetxvec{\pgfpoint{0.80mm}{0mm}}
\pgfsetyvec{\pgfpoint{0mm}{0.80mm}}
\color[rgb]{0,0,0}\pgfsetlinewidth{0.30mm}\pgfsetdash{}{0mm}
\pgfcircle[fill]{\pgfxy(0.00,0.00)}{0.48mm}
\pgfcircle[stroke]{\pgfxy(0.00,0.00)}{0.48mm}
\pgfcircle[fill]{\pgfxy(-5.00,5.00)}{0.48mm}
\pgfcircle[stroke]{\pgfxy(-5.00,5.00)}{0.48mm}
\pgfcircle[fill]{\pgfxy(-10.00,10.00)}{0.48mm}
\pgfcircle[stroke]{\pgfxy(-10.00,10.00)}{0.48mm}
\pgfmoveto{\pgfxy(-10.00,10.00)}\pgflineto{\pgfxy(-5.00,5.00)}\pgfstroke
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\pgfputat{\pgfxy(-2.00,-2.00)}{\pgfbox[bottom,left]{\fontsize{9.10}{10.93}\selectfont \makebox[0pt][r]{2}}}
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\pgfputat{\pgfxy(-12.00,9.00)}{\pgfbox[bottom,left]{\fontsize{9.10}{10.93}\selectfont \makebox[0pt][r]{6}}}
\pgfputat{\pgfxy(-7.50,-10.00)}{\pgfbox[bottom,left]{\fontsize{11.38}{13.66}\selectfont \makebox[0pt]{$T^{(4)}$}}}
\end{pgfpicture}

&&

\centering
\begin{pgfpicture}{-13.20mm}{-10.86mm}{13.20mm}{11.71mm}
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\pgfmoveto{\pgfxy(0.00,0.00)}\pgflineto{\pgfxy(10.00,5.00)}\pgfstroke
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\pgfputat{\pgfxy(-7.00,3.00)}{\pgfbox[bottom,left]{\fontsize{9.10}{10.93}\selectfont \makebox[0pt][r]{2}}}
\pgfputat{\pgfxy(-12.00,9.00)}{\pgfbox[bottom,left]{\fontsize{9.10}{10.93}\selectfont \makebox[0pt][r]{9}}}
\pgfputat{\pgfxy(12.00,4.00)}{\pgfbox[bottom,left]{\fontsize{9.10}{10.93}\selectfont 3}}
\pgfputat{\pgfxy(3.00,9.00)}{\pgfbox[bottom,left]{\fontsize{9.10}{10.93}\selectfont \makebox[0pt][r]{6}}}
\pgfputat{\pgfxy(-2.50,-10.00)}{\pgfbox[bottom,left]{\fontsize{11.38}{13.66}\selectfont \makebox[0pt]{$T^{(3)}$}}}
\end{pgfpicture}

&&

\centering
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\end{align*}
with
\begin{align*}
a^{(4)}&=3, & a^{(3)}&=2, & a^{(2)}&=9, & a^{(1)}&=5,\\
b^{(4)}&=3, & b^{(3)}&=2, & b^{(2)}&\text{ does not exist}, & b^{(1)}&=5.
\end{align*}
Thus, the increasing 1-2 tree $T^{(1)}$ obtained from $\sigma$ under $\psi'$ is the same with the tree obtained from $\sigma$ under $\psi$ in Example 3.2 and Fig. 2 in \cite{GSZ11}.

\end{ex}

\begin{thm}
The two bijections $\psi$ and $\psi'$ from $\Ank$ to $\Tnk$ are equal.
\end{thm}
\begin{proof}

It is clear that (C2) is equivalent to (B1).
Since the rule (B2a) just exchange two labels, but does not change the tree-structure,
it is enough to show that (C1) is produced recursively from (B1) and (B2b).

Assume that $\sigma_{2i-1} > a^{(i)} > \sigma_{2i}$ for some $1 \le i \le m-1$.
We obtain the right tree in the following by applying (B1) to the tree $T^{(i+1)}$.
$$

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\end{pgfpicture}

$$
Due to $\sigma_{2i-1} > v_1$ $(= a^{(i)})$,
it is not an increasing 1-2 tree
and we apply (B2b) to the above tree as follows.
$$

\centering
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\end{pgfpicture}

$$
Since $\sigma_{2i-1} > v_j > \dots > v_2$, until we have an increasing 1-2 tree, repeat to apply (B2b) as follows.
$$

\centering
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\pgfputat{\pgfxy(-112.00,27.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt][r]{$a^{(i)}$}}}
\pgfmoveto{\pgfxy(-100.00,20.00)}\pgfcurveto{\pgfxy(-96.71,22.07)}{\pgfxy(-93.37,24.07)}{\pgfxy(-90.00,26.00)}\pgfcurveto{\pgfxy(-85.85,28.38)}{\pgfxy(-81.61,30.67)}{\pgfxy(-77.00,32.00)}\pgfcurveto{\pgfxy(-73.68,32.96)}{\pgfxy(-70.00,34.27)}{\pgfxy(-70.00,37.50)}\pgfcurveto{\pgfxy(-70.00,39.99)}{\pgfxy(-72.38,41.65)}{\pgfxy(-75.00,42.00)}\pgfcurveto{\pgfxy(-78.11,42.42)}{\pgfxy(-81.15,41.30)}{\pgfxy(-84.00,40.00)}\pgfcurveto{\pgfxy(-89.08,37.69)}{\pgfxy(-93.95,34.68)}{\pgfxy(-97.00,30.00)}\pgfcurveto{\pgfxy(-98.94,27.02)}{\pgfxy(-99.98,23.55)}{\pgfxy(-100.00,20.00)}\pgfstroke
\pgfputat{\pgfxy(-74.00,37.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt][r]{$S_1$}}}
\pgfmoveto{\pgfxy(-100.00,20.00)}\pgflineto{\pgfxy(-90.00,30.00)}\pgfstroke
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\pgfcircle[stroke]{\pgfxy(-105.00,49.00)}{0.00mm}
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\pgfcircle[stroke]{\pgfxy(-70.00,70.00)}{0.55mm}
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\pgfputat{\pgfxy(-70.00,65.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont $b^{(i)}$}}
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\pgfsetdash{{0.60mm}{0.50mm}}{0mm}\pgfsetlinewidth{0.60mm}\pgfmoveto{\pgfxy(-100.00,50.00)}\pgflineto{\pgfxy(-90.00,60.00)}\pgfstroke
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\pgfsetdash{}{0mm}\pgfsetlinewidth{0.30mm}\pgfcircle[stroke]{\pgfxy(-80.00,60.00)}{0.55mm}
\pgfmoveto{\pgfxy(-90.00,70.00)}\pgflineto{\pgfxy(-80.00,60.00)}\pgfstroke
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\pgfcircle[stroke]{\pgfxy(-90.00,70.00)}{0.55mm}
\pgfputat{\pgfxy(-106.00,76.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont $S_{j-1}$}}
\pgfputat{\pgfxy(-126.00,56.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont $S_2$}}
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\pgfputat{\pgfxy(-110.00,52.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt][r]{$u_2$}}}
\pgfputat{\pgfxy(-80.00,55.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont $v_{j-1}$}}
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\pgfsetlinewidth{0.90mm}\pgfmoveto{\pgfxy(-110.00,30.00)}\pgflineto{\pgfxy(-70.00,70.00)}\pgfstroke
\pgfputat{\pgfxy(-101.00,35.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont $v_2$}}
\pgfsetlinewidth{0.30mm}\pgfmoveto{\pgfxy(-90.00,10.00)}\pgflineto{\pgfxy(-110.00,30.00)}\pgfstroke
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\pgfcircle[stroke]{\pgfxy(-120.00,40.00)}{0.55mm}
\pgfcircle[fill]{\pgfxy(-100.00,20.00)}{0.55mm}
\pgfcircle[stroke]{\pgfxy(-100.00,20.00)}{0.55mm}
\pgfsetlinewidth{1.20mm}\pgfmoveto{\pgfxy(-40.00,30.00)}\pgflineto{\pgfxy(-30.00,30.00)}\pgfstroke
\pgfmoveto{\pgfxy(-30.00,30.00)}\pgflineto{\pgfxy(-32.80,30.70)}\pgflineto{\pgfxy(-32.80,29.30)}\pgflineto{\pgfxy(-30.00,30.00)}\pgfclosepath\pgffill
\pgfmoveto{\pgfxy(-30.00,30.00)}\pgflineto{\pgfxy(-32.80,30.70)}\pgflineto{\pgfxy(-32.80,29.30)}\pgflineto{\pgfxy(-30.00,30.00)}\pgfclosepath\pgfstroke
\pgfsetdash{{0.60mm}{0.50mm}}{0mm}\pgfsetlinewidth{0.60mm}\pgfmoveto{\pgfxy(-23.00,30.00)}\pgflineto{\pgfxy(-13.00,30.00)}\pgfstroke
\pgfputat{\pgfxy(-35.00,37.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt]{(B2b)}}}
\pgfputat{\pgfxy(-5.00,37.00)}{\pgfbox[bottom,left]{\fontsize{6.26}{7.51}\selectfont \makebox[0pt]{(B2b)}}}
\end{pgfpicture}

$$
Since (C2a) is produced from the rule (B1) and (B2b),
then Algorithm~C follows Algorithm~B.
\end{proof}

\section*{Acknowledgements}
The first author's work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2017R1C1B2008269).