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Consider a sequence of integers $n_i,\ i=1,\ldots, N$ and $\nu_k=\sum_{i=k}^N n_i$. Consider a sequence $\Delta_i,\ i=0,\ldots, N+1$ with $\Delta_i\in \{0,1\}$ and $\Delta_0=\Delta_{N+1}=0$. For $i=0,\ldots,N-1$, consider $$\epsilon_i=\frac{n_i+\Delta_i-\Delta_{i-1}}{2}+\partial_i-\partial_{i+1},\qquad \epsilon_N=\frac{n_N-\Delta_{N-1}}{2}+\partial_N$$ where $$\partial_i=\left\{ \begin{array}{cc} \Delta_{i} & if\ \nu_{i}-\Delta_{i-1}\ is\ even \\ \frac{1}{2} & else \end{array}\right.$$ Show that there exists a unique sequence $\Delta_i$ such that

  • if $\Delta_i=1$ then $\epsilon_i\geq n_i$

  • else $\epsilon_i \geq 2-\Delta_{i-1}-\Delta_{i+1}$

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    $\begingroup$ Why do you say "show", not ask whether this is true or not? Is there a strong evidence? $\endgroup$
    – Fedor Petrov
    Commented Oct 1, 2020 at 17:37
  • $\begingroup$ Well, I am pretty convinced that this is true : I have an implemented an algorithm that computes the correct sequence of $\Delta_i$. This algorithm never failed. The most strange fact, at least to me, is the uniqueness. Notice that if, for instance, the integer $n_i$ is bigger than $4$ then the only possibility is $\Delta_i=0$. Thus, the problem reduces to small values of $n_i$. One last comment : this question comes from some consideration in geometry of germ of complex curve in the complex plane. $\endgroup$
    – Yoyo
    Commented Oct 2, 2020 at 13:23
  • $\begingroup$ you mean $\nu_k=\sum_{i=k}^N n_i$? $\endgroup$
    – Fedor Petrov
    Commented Oct 2, 2020 at 21:10
  • $\begingroup$ Yes, you are right. I corrected. $\endgroup$
    – Yoyo
    Commented Oct 4, 2020 at 8:28

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