Euler characteristic for a singular-complex of a manifold is a topological invariant, since it does not change under re-triangularization of the singular-complex. It is also a linear combination of numbers of simplexes at each dimensions.
My question is that is Euler characteristic the only linear combination of numbers of simplexes that is invariant under re-triangularization of the singular-complex?
This question may be difficult to formulate precisely. If you allow the coefficients of the linear combination to depend on the topology of the space, then you can trivially make such linear combinations. So maybe you are asking whether there is a universal set of coefficients $n_i$ such that $\sum n_i (\text{# of $i$-simplices})$ is a toplogical invariant? I suspect that the Euler characteristic is the unique answer up to scalar multiples, which you can surmise by plugging in various simplicial complexes to get constraints on the coefficients $n_i$. For example, subdividing an interval into $k$ subintervals we get that $n_0\cdot(k+1)+n_1\cdot(k)$ must be constant. This implies $n_1=-n_0$. Looking at subdivisions of successively higher dimensional simplices should prove that $n_i=(-1)^{i}n_0$, which gives your invariant as $n_0\cdot \chi$.