Here is an interesting inequality that is simulated to be correct but I can not prove it. Can someone helps me?
The question is that: For any binary vector $\mathbf{y}\in\{0,1\}^n$, prove that \begin{equation} \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0\leq k, H(\mathbf{x},\mathbf{y})\leq d\} \leq \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0\leq k, H(\mathbf{x},\mathbf{1})\leq d\} \end{equation} holds for all $0\leq k \leq n-1$ and $0\leq d<\frac{n}{2}$, where $\nabla(\cdot)$ denotes the first-order difference operator (i.e., $\nabla(\mathbf{x})=(\mathbf{x}_2-\mathbf{x}_1,\mathbf{x}_3-\mathbf{x}_2,\cdots,\mathbf{x}_n-\mathbf{x}_{n-1})$), $H(\cdot,\cdot)$ denotes the Hamming distance, and $\mathbf{1}=(1,1,\cdots,1)^\top$ is the all-ones vector.
Remark 1: the condition $d<\frac{n}{2}$ is necessary, otherwise, if $d\geq \frac{n}{2}$, the simulation experiment can find the counter-example and thus the inequality does not hold.
Remark 2: the above inequality can not be sepearted for each $k,d$ case, i.e., the following inequalities $$ \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0= k, H(\mathbf{x},\mathbf{y})= d\} \leq \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0= k, H(\mathbf{x},\mathbf{1})= d\}, $$ $$ \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0= k, H(\mathbf{x},\mathbf{y})\leq d\} \leq \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0= k, H(\mathbf{x},\mathbf{1})\leq d\}, $$ and $$ \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0\leq k, H(\mathbf{x},\mathbf{y})= d\} \leq \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0\leq k, H(\mathbf{x},\mathbf{1})= d\} $$ are false that are verified by simulation experiments.
Remark 3: it is easy to see that $\#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0\leq k, H(\mathbf{x},\mathbf{1})\leq d\}=\#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0\leq k, H(\mathbf{x},\mathbf{0})\leq d\}$, where $\mathbf{0}=(0,0,\cdots,0)^\top$ denotes the all-zeros vector. This does not affect the correctness of the above inequality.
Remark 4: an alternative inequality is that, for any $\mathbf{y}\in\{0,1\}^n$, prove that $$ \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0\leq k, H(\mathbf{x},\mathbf{y})\leq d\} \leq \#\{\mathbf{x}\in\{0,1\}^n \mid \|\nabla(\mathbf{x})\|_0\leq k, H(\mathbf{x},\mathbf{z})\leq d\} $$ holds for all $0\leq k\leq n-1$ and $0\leq d<\frac{n}{2}$, where $\mathbf{z}=(1,1,\cdots,1,0,\cdots,0)$ satisfying $\|\mathbf{z}\|_0=\|\mathbf{y}\|_0$. This inequality is also verified to be true in simulation experiments. If one can prove thus inequality, I can then prove the first inequality to be ture.
Remark 5: I have tried a lot efforts to prove this but all failed. E.g., I tried to construct a monotonic mapping from the left set to the right set, but failed! I tried to prove this inequality by using the mathematical induction, but failed!
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