# construct a bijective map between subsets of binary sequence

Consider the binary sequence $$\{0,1\}^N$$ where $$N$$ is an even integer (for simplicity). Let $$M_k := \{\beta\in \{0,1\}^N \rvert \sum_{j=1}^N \beta_j = k\}$$ (i.e., $$M_k$$ is the set that contains all binary sequences with length $$N$$ and with exactly $$k$$ 1's). The question is: for general $$k \le N/2$$, is it always possible to construct a bijective map $$\phi: M_k\rightarrow M_{N-k}$$ such that $$\phi(\beta)_j \ge \beta_j$$? (Here, $$\phi(\beta)_j$$ is the $$j^{th}$$ coordinate of the binary sequence $$\phi(\beta)$$.)

Obviously, both sets $$M_k$$ and $$M_{N-k}$$ have the same number of elements. When $$k = 1$$, the construction of $$\phi$$ is simple. However, for general $$k$$ and $$N$$, I am not sure how to systematically construct such a function $$\phi$$. Any hint or reference would also help a lot. Thank you.

The answer is "yes, this is always possible." The bijection on ranks $$k$$ and $$N-k$$ induced from a symmetric chain decomposition of the Boolean lattice gives you what you're asking for. For basics on this topic, see these slides.