On the Livingstone-Wagner Theorem - Mnukhin, V. B.; Siemons, I. J. (The Electronic Journal of Combinatorics, 2004)
Let $G$ be a permutation group on the set $Ω$ and let ${\cal S}$ be a collection of subsets of $Ω,$ all of size $≥ m$ for some integer $m$. For $s≤ m$ let $n_{s}(G,\,{\cal S})$ be the number of $G$-orbits on the subsets of $Ω$ which have a representative $y\subseteq x$ with $|y|=s$ and $y\subseteq x$ for some $x\in {\cal S}$. We prove that if $s < t$ with $s+t≤ m$ then $n_{s}(G,\,{\cal S})≤ n_{t}(G,\,{\cal S})$. A special case of this theorem is the Livingstone-Wagner Theorem when ${\cal S}=\{Ω\}$. We show how the result can be applied to estimate orbit numbers for simplicial complexes, sequences, graphs and amalgamation classes. It is also shown how this theorem can be extended to orbit theorems on more general partially ordered sets.