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Combinatorial Inequality

Consider a set of $2^n-1$ non negative integers $S= ${$ a_{i,j}|1\le i\le n; 1\le j\le 2^{i-1} $} such that: \begin{align}{} 1.\ \ &a_{i,j}\le 2^{n+1-i} \\\\ 2.\ \ &a_{i,j}\le a_{i-1,j} \\\\ ...