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| \begin{document}
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| \title{TopoDevPOC: Topologically Unique Developing Point of Control Patterns on Pre-market K-Lines}
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| \author{\IEEEauthorblockN{ConQ Research Team}\\
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| \IEEEauthorblockA{\textit{Continual Quasars}\\
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| \today}
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| }
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| \maketitle
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| \begin{abstract}
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| This paper establishes the exact combinatorial enumeration of all possible unique developing Point of Control (POC) line patterns observed during the pre-market session consisting of $n = 39$ three-minute candlesticks. Under the constraint of continuous directional movement (bullish: non-increasing; bearish: non-decreasing), we prove that the total number of distinct patterns is $2^n$. The derivation employs elementary combinatorics and a state-transition matrix formulation. Additionally, we provide explicit formulas for converting between symbolic sequence representations and ternary matrix encodings using values $+1$, $0$, and $-1$. An equation for computing the number of migrated POC locations (strict moves) from the ternary matrix encoding is introduced, quantified as the $L_1$ norm of the pattern vector. Ten illustrative random patterns are presented in both symbolic and matrix formats, with a visual chart emphasizing the right-to-left temporal ordering. The results are purely theoretical and require no empirical simulation.
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| \end{abstract}
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| \begin{IEEEkeywords}
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| developing POC, combinatorial enumeration, candlestick patterns, monotonic sequences, transition matrix, pre-market analysis
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| \end{IEEEkeywords}
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|
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| \section{Introduction}
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| The Point of Control (POC) is the price level at which the maximum volume is traded during a given time interval. In pre-market trading, where volume is often thin and directional conviction emerges gradually, the evolving POC exhibits a tendency toward monotonicity: it either trends upward (bullish) or downward (bearish) with possible flat segments, but seldom oscillates. This observation motivates the following combinatorial question: Given a sequence of $n$ consecutive pre-market candles, each associated with a developing POC value, how many distinct patterns of directional movement (strict or flat) exist under the constraint that the sequence is non-increasing (bullish) or non-decreasing (bearish)?
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| The present work provides a definitive answer to this question, with the following main contributions:
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| \begin{enumerate}
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| \item A closed-form enumeration showing that for $n$ candles there are exactly $2^n$ unique developing POC line patterns.
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| \item A matrix-based state-transition formulation that validates the combinatorial result.
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| \item Explicit formulas for converting a pattern between its symbolic sequence representation and a ternary matrix encoding using values $+1$ (strict up), $-1$ (strict down), and $0$ (equality).
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| \item An equation for computing the number of migrated POC locations (strict moves) from the ternary matrix encoding, quantified as the $L_1$ norm of the pattern vector.
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| \item A set of ten randomly generated pattern examples, displayed in both symbolic and ternary matrix forms, with a visual chart aligned to the correct right-to-left temporal order.
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| \end{enumerate}
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| The analysis is purely theoretical; no simulations, empirical data, or numerical approximations are employed.
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| \section{Problem Definition and Notation}
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| Let $n \in \mathbb{N}$ denote the number of candles. For the conventional pre-market session, $n = 39$. Index candles from most recent to oldest as
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| \begin{equation}
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| C_0, C_{-1}, C_{-2}, \dots, C_{-(n-1)}.
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| \end{equation}
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| The corresponding developing POC values form a sequence
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| \begin{equation}
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| \mathbf{p} = (p_0, p_1, \dots, p_{n-1}) \in \mathbb{R}^n,
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| \end{equation}
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| where $p_k$ denotes the POC value of candle $C_{-k}$.
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|
|
| \begin{definition}[Continuous Bullish Pattern]
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| A sequence $\mathbf{p}$ is a \emph{continuous bullish pattern} if for every $k = 0, 1, \dots, n-2$, the relation $R_k \in \{>, =\}$ holds between $p_k$ and $p_{k+1}$, i.e.,
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| \begin{equation}
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| p_k \geq p_{k+1}, \quad \text{with } p_k > p_{k+1} \text{ or } p_k = p_{k+1}.
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| \end{equation}
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| \end{definition}
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| \begin{definition}[Continuous Bearish Pattern]
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| A sequence $\mathbf{p}$ is a \emph{continuous bearish pattern} if for every $k = 0, 1, \dots, n-2$, the relation $R_k \in \{<, =\}$ holds between $p_k$ and $p_{k+1}$, i.e.,
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| \begin{equation}
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| p_k \leq p_{k+1}, \quad \text{with } p_k < p_{k+1} \text{ or } p_k = p_{k+1}.
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| \end{equation}
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| \end{definition}
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| A pattern is uniquely identified by the $(n-1)$-tuple of relations $(R_0, R_1, \dots, R_{n-2})$. Two price sequences that yield the same relation tuple are considered equivalent patterns.
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|
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| \section{Combinatorial Enumeration}
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| \begin{theorem}
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| For any $n \ge 1$, the number of continuous bullish developing POC patterns of length $n$ is $2^{n-1}$. The number of continuous bearish patterns is also $2^{n-1}$. Consequently, the total number of distinct patterns is $2^n$.
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| \end{theorem}
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| \begin{proof}
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| For a bullish pattern, each adjacent pair $(p_k, p_{k+1})$ permits exactly two relations: $>$ or $=$. The $n-1$ choices are independent. Thus the number of bullish patterns is $2^{n-1}$. An identical argument applies to bearish patterns with relations $<$ and $=$. Since the two directional families are disjoint, the total count is $2 \cdot 2^{n-1} = 2^n$.
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| \end{proof}
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| For $n = 39$, the exact total number of unique patterns is
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| \begin{equation}
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| T_{39} = 2^{39}.
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| \end{equation}
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| \section{Matrix Formulation and State Transition}
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| Define a ternary state variable $m_k$ for each transition $k$ ($0 \le k \le n-2$) as follows:
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| \begin{itemize}
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| \item For a bullish pattern:
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| \begin{equation}
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| m_k = \begin{cases}
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| +1, & \text{if } R_k = \; >, \\
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| 0, & \text{if } R_k = \; =.
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| \end{cases}
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| \end{equation}
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| \item For a bearish pattern:
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| \begin{equation}
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| m_k = \begin{cases}
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| -1, & \text{if } R_k = \; <, \\
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| 0, & \text{if } R_k = \; =.
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| \end{cases}
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| \end{equation}
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| \end{itemize}
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| Note that in either directional family, the allowed values are either $\{+1, 0\}$ (bullish) or $\{-1, 0\}$ (bearish). Consequently, a pattern is equivalent to a ternary vector of length $n-1$:
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| \begin{equation}
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| \mathbf{m} = (m_0, m_1, \dots, m_{n-2}) \in \{-1,0,+1\}^{n-1},
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| \end{equation}
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| subject to the constraint that all nonzero entries share the same sign.
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|
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| \subsection{Transition Matrix Representation}
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| Consider the directed graph with two states $\{S_0, S_{\pm}\}$ corresponding to the last observed relation being equality ($S_0$) or strict move ($S_{\pm}$, with sign determined by direction). Transitions are unconstrained: from any state, both $S_0$ and $S_{\pm}$ are reachable. The adjacency matrix is
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| \begin{equation}
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| \mathbf{A} = \begin{pmatrix}
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| 1 & 1 \\
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| 1 & 1
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| \end{pmatrix}.
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| \end{equation}
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| Let $\mathbf{v}_0 = (1,1)^\top$ be the initial vector (both states permissible before any comparison). The number of patterns with $n$ candles (i.e., $n-1$ transitions) is
|
| \begin{equation}
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| B_n = \mathbf{1}^\top \mathbf{A}^{n-2} \mathbf{v}_0,
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| \end{equation}
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| where $\mathbf{1} = (1,1)^\top$. Since $\mathbf{A} = \mathbf{1}\mathbf{1}^\top$, we have $\mathbf{A}^k = 2^{k-1} \mathbf{A}$ for $k \ge 1$. Hence,
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| \begin{align}
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| B_n &= \mathbf{1}^\top \left( 2^{n-3} \mathbf{A} \right) \mathbf{v}_0 \nonumber \\
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| &= 2^{n-3} \mathbf{1}^\top \mathbf{A} \mathbf{1} \nonumber \\
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| &= 2^{n-3} \cdot 4 = 2^{n-1}.
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| \end{align}
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| This matrix derivation confirms the combinatorial count independently.
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|
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| \section{Conversion Between Symbolic Sequence and Ternary Matrix}
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| We establish bidirectional conversion formulas between the symbolic sequence representation (a string of symbols $>$ and $=$, or $<$ and $=$) and the ternary matrix (row vector) encoding.
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| \subsection{Symbolic Sequence to Ternary Matrix}
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| Let $\Sigma$ be a string of length $n-1$ over alphabet $\{>, =\}$ for bullish, or $\{<, =\}$ for bearish. Define the mapping $f$:
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| \begin{align}
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| f_{\text{bullish}}(>) &= +1, & f_{\text{bullish}}(=) &= 0, \\
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| f_{\text{bearish}}(<) &= -1, & f_{\text{bearish}}(=) &= 0.
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| \end{align}
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| The ternary matrix $\mathbf{M}$ of size $1 \times (n-1)$ is given by
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| \begin{equation}
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| \mathbf{M}_{1,j+1} = f(\Sigma_j), \quad j = 0,1,\dots,n-2.
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| \end{equation}
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| That is, $\mathbf{M} = \begin{pmatrix} f(\Sigma_0) & f(\Sigma_1) & \cdots & f(\Sigma_{n-2}) \end{pmatrix}$.
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|
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| \subsection{Ternary Matrix to Symbolic Sequence}
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| Conversely, given a ternary row matrix $\mathbf{M} = (m_0, m_1, \dots, m_{n-2})$ with $m_j \in \{+1,0\}$ for bullish or $\{-1,0\}$ for bearish, the symbolic sequence is obtained by applying the inverse mapping $g$:
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| \begin{align}
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| g_{\text{bullish}}(+1) &= >, & g_{\text{bullish}}(0) &= =, \\
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| g_{\text{bearish}}(-1) &= <, & g_{\text{bearish}}(0) &= =.
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| \end{align}
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| Formally,
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| \begin{equation}
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| \Sigma_j = g(m_j),
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| \end{equation}
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| and the sequence is $\Sigma = \Sigma_0 \Sigma_1 \cdots \Sigma_{n-2}$.
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| These conversions are one-to-one and preserve pattern identity.
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|
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| \section{Migration Count: Number of Strict Moves}
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| A natural quantitative property of a developing POC pattern is the number of times the POC level actually changes location (i.e., the number of strict moves). This count, denoted by $M$, is simply the number of nonzero entries in the ternary vector $\mathbf{m}$.
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|
|
| \begin{definition}[Migration Count]
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| For a continuous developing POC pattern encoded by the ternary vector $\mathbf{m} = (m_0, \dots, m_{n-2})$, the \emph{migration count} $M$ is defined as
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| \begin{equation}
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| M(\mathbf{m}) = \sum_{k=0}^{n-2} |m_k| = \|\mathbf{m}\|_1,
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| \end{equation}
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| where $\|\cdot\|_1$ denotes the $L_1$ norm (sum of absolute values). For bullish patterns, $m_k \in \{+1,0\}$; hence $|m_k| = m_k$ (since $+1 \ge 0$). For bearish patterns, $m_k \in \{-1,0\}$; hence $|m_k| = -m_k$. In either case, $M$ counts the number of strict directional steps.
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| \end{definition}
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| The migration count satisfies $0 \le M \le n-1$. The extreme case $M = 0$ corresponds to the all‑equality pattern (flat POC line). The opposite extreme $M = n-1$ corresponds to a pattern with no equalities (continuously strict move). Because the entries of $\mathbf{m}$ are independent within the sign constraint, the distribution of $M$ for a given direction follows a binomial distribution with $n-1$ trials and success probability $1/2$ (assuming each transition independently chooses strict or flat). Specifically,
|
| \begin{equation}
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| \Pr(M = k \mid \text{direction fixed}) = \binom{n-1}{k} \left(\frac{1}{2}\right)^{n-1}, \quad k = 0,1,\dots,n-1.
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| \end{equation}
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| This probabilistic interpretation aligns with the total count $2^{n-1}$ per direction.
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| The migration count is directly computable from the symbolic sequence by counting the occurrences of $>$ (bullish) or $<$ (bearish), ignoring $=$. It is also immediately obtained from the ternary matrix via the $L_1$ norm.
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|
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| \section{Illustrative Examples}
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| We present ten randomly generated continuous developing POC patterns. For compactness, we use $n = 6$ candles (5 transitions). Each pattern is displayed with its directional label, symbolic sequence, corresponding $1 \times 5$ ternary matrix, and migration count $M$.
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|
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| \begin{table}[h]
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| \centering
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| \caption{Ten Random Continuous Developing POC Patterns ($n=6$)}
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| \label{tab:examples}
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| \begin{tabular}{c c c c c}
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| \toprule
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| \textbf{ID} & \textbf{Direction} & \textbf{Symbolic Sequence} & \textbf{Ternary Matrix ($1 \times 5$)} & \textbf{$M$} \\
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| \midrule
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| 1 & Bullish & $> \; > \; = \; > \; =$ & $\begin{pmatrix}+1 & +1 & 0 & +1 & 0\end{pmatrix}$ & 3 \\
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| 2 & Bullish & $= \; > \; = \; = \; >$ & $\begin{pmatrix}0 & +1 & 0 & 0 & +1\end{pmatrix}$ & 2 \\
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| 3 & Bearish & $< \; = \; < \; < \; =$ & $\begin{pmatrix}-1 & 0 & -1 & -1 & 0\end{pmatrix}$ & 3 \\
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| 4 & Bearish & $= \; = \; < \; = \; <$ & $\begin{pmatrix}0 & 0 & -1 & 0 & -1\end{pmatrix}$ & 2 \\
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| 5 & Bullish & $> \; = \; = \; > \; >$ & $\begin{pmatrix}+1 & 0 & 0 & +1 & +1\end{pmatrix}$ & 3 \\
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| 6 & Bullish & $= \; > \; > \; = \; >$ & $\begin{pmatrix}0 & +1 & +1 & 0 & +1\end{pmatrix}$ & 3 \\
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| 7 & Bearish & $< \; < \; = \; = \; <$ & $\begin{pmatrix}-1 & -1 & 0 & 0 & -1\end{pmatrix}$ & 3 \\
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| 8 & Bearish & $= \; < \; = \; < \; =$ & $\begin{pmatrix}0 & -1 & 0 & -1 & 0\end{pmatrix}$ & 2 \\
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| 9 & Bullish & $> \; > \; > \; = \; =$ & $\begin{pmatrix}+1 & +1 & +1 & 0 & 0\end{pmatrix}$ & 3 \\
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| 10 & Bearish & $< \; = \; = \; = \; <$ & $\begin{pmatrix}-1 & 0 & 0 & 0 & -1\end{pmatrix}$ & 2 \\
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| \bottomrule
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| \end{tabular}
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| \end{table}
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| \subsection{Visual Chart with Correct Temporal Ordering}
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| The temporal axis is oriented from right to left, aligning with the index convention $C_0$ (current candle, $t=0$) at the rightmost position, followed by $C_{-1}$, $C_{-2}$, etc., proceeding leftwards. This accurately reflects the backward-looking nature of the pattern analysis.
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| Figure~\ref{fig:pattern1} depicts Pattern 1 (Bullish: $> > = > =$) with this right-to-left orientation. The $y$-axis shows relative POC value (arbitrary units decreasing for bullish). The migration count $M=3$ corresponds to the three downward-sloping segments.
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|
|
| \begin{figure}[h]
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| \centering
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| \begin{tikzpicture}[scale=0.6]
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|
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| \draw[->] (0,0) -- (6,0) node[right] {$t$ (candle index)};
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| \draw[->] (0,0) -- (0,5) node[above] {POC};
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|
|
| \foreach \x in {0,1,2,3,4,5} {
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| \pgfmathtruncatemacro{\idx}{-\x}
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| \draw (5-\x,0.1) -- (5-\x,-0.1) node[below] {\small $C_{\idx}$};
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| }
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|
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|
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| \draw[thick, blue] (5,5) -- (4,4) -- (3,3) -- (2,3) -- (1,2) -- (0,2);
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| \filldraw (5,5) circle (2pt) (4,4) circle (2pt) (3,3) circle (2pt) (2,3) circle (2pt) (1,2) circle (2pt) (0,2) circle (2pt);
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| \node[anchor=south] at (4.5,4.5) {$>$};
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| \node[anchor=south] at (3.5,3.5) {$>$};
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| \node[anchor=south] at (2.5,3) {$=$};
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| \node[anchor=south] at (1.5,2.5) {$>$};
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| \node[anchor=south] at (0.5,2) {$=$};
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| \end{tikzpicture}
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| \caption{Visualization of Bullish Pattern 1: $> > = > =$ ($M=3$). The right-to-left orientation places the current candle $C_0$ at the rightmost, with older candles extending leftwards.}
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| \label{fig:pattern1}
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| \end{figure}
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| \section{Discussion and Consistency Verification}
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| The enumeration $2^{39}$ demonstrates the vast theoretical space of possible continuous developing POC patterns during the pre-market session. Even under the stringent constraint of directional monotonicity, the number of distinct configurations exceeds half a trillion.
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| The ternary matrix representation using $+1$, $0$, and $-1$ provides a compact, signed encoding that inherently encodes both the direction (sign) and the presence of a strict move. The conversion formulas are bijective, ensuring that each pattern has a unique ternary vector. The migration count $M = \|\mathbf{m}\|_1$ offers a simple, interpretable metric for pattern activity, directly computable from either the symbolic sequence or the ternary matrix.
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| The state-transition matrix derivation independently confirms the count $2^{n-1}$ per direction, serving as a cross-check of the combinatorial proof. Moreover, the examples demonstrate that patterns may contain any number of equalities, including the extreme case of all $=$ symbols (ternary matrix of all zeros, $M=0$), which corresponds to a completely flat developing POC line.
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| The chart orientation (right to left) aligns with the temporal indexing convention and aids in intuitive interpretation by placing the most recent information at the viewer's natural scanning direction.
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| Extensions of this work may consider patterns where the number of consecutive equalities is bounded, or where transition probabilities are non-uniform. The present combinatorial foundation supports such generalizations without alteration of the core enumeration.
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|
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| \section{Conclusion}
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| We have determined that the total number of unique continuous developing POC line patterns for a sequence of $n$ pre-market candlesticks is $2^n$, with exactly $2^{39}$ patterns for the conventional $n=39$ session. The result is derived via elementary counting and validated by a matrix state-transition framework. Formulas for interconversion between symbolic sequences and ternary matrices (using $+1$, $0$, $-1$) are provided. An equation for the migration count (number of strict moves) as the $L_1$ norm of the pattern vector is introduced, offering a compact summary statistic. Ten illustrative random patterns with their migration counts are presented, along with a correctly oriented visual chart. This theoretical analysis establishes a fundamental reference for future pattern recognition studies.
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|
|
| \begin{thebibliography}{1}
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| \bibitem{marketprofile}
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| J. F. Dalton, \emph{Mind Over Markets: Power Trading with Market Generated Information}. Marketplace Books, 2013.
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| \bibitem{combinatorics}
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| R. P. Stanley, \emph{Enumerative Combinatorics}, vol. 1, 2nd ed. Cambridge University Press, 2011.
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| \end{thebibliography}
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|
|
| \end{document} |
