Flips in symmetric separated set-systems

Abstract

For a positive integer $n$, a collection $S$ of subsets of $[n]=\{1,\ldots,n\}$ is called symmetric if $X\in S$ implies $X^\ast\in S$, where $X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$ (the involution $\ast$ was introduced by Karpman). Leclerc and Zelevinsky showed that the set of maximal strongly (resp. weakly) separated collections in $2^{[n]}$ is connected via flips, or mutations, ``in the presence of six (resp. four) witnesses''. We give a symmetric analog of those results, by showing that each maximal symmetric strongly (weakly) separated collection in $2^{[n]}$ can be obtained from any other one by a series of special symmetric local transformations, so-called symmetric flips. Also we establish the connectedness via symmetric flips for the class of maximal symmetric $r$-separated collections in $2^{[n]}$ when $n,r$ are even (where sets $A,B\subseteq [n]$ are called $r$-separated if there are no elements $i_0<i_1< \cdots <i_{r+1}$ in $[n]$ which alternate in $A\setminus B$ and $B\setminus A$). This is related to a symmetric version of higher Bruhat orders. These results are obtained as consequences of our study of related geometric objects: symmetric rhombus and combined tilings and symmetric cubillages.