Simplicial and combinatorial versions of higher symmetric topological complexity

Abstract

In this paper, we introduce higher symmetric simplicial complexity $SC_n^{\Sigma}(K)$ of a simplicial complex $K$ and higher symmetric combinatorial complexity $CC_n^{\Sigma}(P)$ of a finite poset $P$. These are simplicial and combinatorial approaches to symmetric motion planning of Basabe - Gonz\'{a}lez - Rudyak - Tamaki. We prove that the symmetric simplicial complexity $SC_n^{\Sigma}(K)$ is equal to symmetric topological complexity $TC_n^{\Sigma}(|K|)$ of the geometric realization of $K$ and the symmetric combinatorial complexity $CC_n^{\Sigma}(P)$ is equal to symmetric topological complexity $TC_n^{\Sigma}(|\mathcal{K}(P)|)$ of the geometric realization of the order complex of $P$.