Characterization of finite metric space by their isometric sequences

Let $(X,d)$ be a finite metric space with $\mathrm{\mid }X\mathrm{\mid }=n$. For a positive integer $k$ we define $A_k(X)$ to be the quotient set of all $k$-subsets of $X$ by isometry, and we denote $\mathrm{\mid }{A}_k(X)\mathrm{\mid }$ by $a_k$. The sequence $(a_1,a_2,\dots ,a_n)$ is called the isometric sequence of $(X,d)$. In this article we aim to characterize finite metric spaces by their isometric sequences under one of the following assumptions: (i) $a_k=1$ for some $k$ with $2\le k\le n-2$; (ii) $a_k=2$ for some $k$ with $4\leq k\leq \frac{1+\sqrt{1+4n}}{2}$;(iii)$a_3=2$;(iv)$a_2=a_3=3$. Furthermore, we give some criterion on how to embed such finite metric spaces to Euclidean spaces. We give some maximum cardinalities of subsets in the $d$-dimensional Euclidean space with small $a_3$, which are analogue problems on a sets with few distinct triangles discussed by Epstein, Lott, Miller and Palsson.