Decompositions of the Boolean lattice into rank-symmetric chains

TL;DR

This paper constructs a bijection and partial order on subsets of [n] to decompose the Boolean lattice into rank-symmetric chains, and applies this to asymptotically address F"uredi's chain partition conjecture.

Contribution

It introduces a novel bijection and partial order on subsets of [n] that enable decomposition into rank-symmetric chains, advancing understanding of lattice decompositions.

Findings

Existence of a bijection and partial order satisfying key properties

Decomposition of the Boolean lattice into rank-symmetric chains

Asymptotic proof of F"uredi's chain partition conjecture with rank-symmetric chains

Abstract

The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion. A chain $c_{0}\subset...\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if $|c_{i}|+|c_{k-i}|=n$ for $i=0,...,k$; and it is symmetric, if $|c_{i}|=(n-k)/2+i$. We show that there exist a bijection $$p: [n]^{(\geq n/2)}\rightarrow [n]^{(\leq n/2)}$$ and a partial ordering $<$ on $[n]^{(\geq n/2)}$ satisfying the following properties: (i) $\subset$ is an extension of $<$ on $[n]^{(\geq n/2)}$; (ii) if $C\subset [n]^{(\geq n/2)}$ is a chain with respect to $<$, then $p(C)\cup C$ is a rank-symmetric chain in $2^{[n]}$, where $p(C)=\{p(x): x\in C\}$; (iii) the poset $([n]^{(\geq n/2)},<)$ has the so called normalized matching property. We show two applications of this result. A conjecture of F\"{u}redi asks if $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that the size of any two chains differ by at most 1. We prove an asymptotic version of this conjecture with the additional condition that every chain in the partition is rank-symmetric: $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ rank-symmetric chains, each of size $\Theta(\sqrt{n})$.