Suppose we have a symmetric polynomial $P$ in $n$ variables. We can partition this alphabet into sets with one or two letters, e.g. ${ {x_1}, {x_2, x_3}}.

We can thus see $P$ as an element in $Q[x_1][x_2,x_3]$, and expand it in the Schur basis in each sub-set of variables. E.g, $P=5 s_{2}(x_1)s_{32}(x_2,x_3)$ or similar.

Suppose now that this expansion always have non-negative coefficients, *for every choice of partition of letters into 1-or-2-element subsets*.

**Can we conclude that $P$ itself is Schur-positive in $n$ variables?**
Note that the Littlewood-Richardson rule tells us that the converse is true.

If not, is there some $k = k(n)$ such that Schur-positivity on subsets of size $\leq k$ implies Schur-positivity in $n$ variables?

The intuition behind why 2-element sets might be enough is as follows: If we want to show that a sum over some combinatorial objects is Schur-positive, it suffices to create a map to SSYTs, or equivalently, to reading-words which are Knuth-equivalent to the SSYTs. To completely describe a reading-word, it suffices to know how many times $i$ appear before $j$, for every pair $i$, $j$.

So, assume that I have a set of objects in bijection with SSYTs of shape $\lambda$. By expanding the sum over these it in different choices of 2-variable subsets, we can almost figure out this data above. But I cannot make this formal, at least without using type A crystals or similar...