The monomial symmetric functions form the integral stable basis

In degree $d=0$, the only partition is $\mathrm{\varnothing }$, its orbit sum is the constant $1$, and ${\mathrm{\Lambda }}^0=\mathbb{Z}$; hence it is a basis.

Suppose $d>0$ and fix $N\ge d$. Every partition of $d$ has at most $d$ parts, so every $\lambda ⊢d$ has $\mathrm{\ell }(\lambda )\le N$. By [F3], the rank-$N$ orbit sums indexed by these partitions form a $\mathbb{Z}$-basis of $A_N^d$.

If $M>N\ge d$, specializing $x_{N+1},\dots ,x_M$ to zero leaves exactly those orbit monomials whose positive exponents all lie among the first $N$ variables; these are precisely the distinct rank-$N$ orbit monomials, each once. Thus every transition $A_M^d\to A_N^d$ is an isomorphism carrying the displayed basis to itself.

A compatible sequence in ${\mathrm{\Lambda }}^d$ is uniquely determined by its rank-$N$ component. Expanding that component in the finite basis of [F3], compatibility and the basis-preserving isomorphisms of step 1.3 force the same integer coefficients at every rank $M\ge N$; lower-rank components are their specializations. Conversely, every finite integer combination of the compatible orbit sums gives such a sequence. Hence the stable orbit sums span and are linearly independent in ${\mathrm{\Lambda }}^d$.