We compute an explicit e-positive formula for the chromatic symmetric function of a lollipop graph, L m,n . From here we deduce that there exist countably infinite distinct e-positive, and hence Schur-positive, bases of the algebra of symmetric functions whose generators are chromatic symmetric functions. Finally, we resolve 6 conjectures on the chromatic symmetric function of a lariat graph, L n+3 .

We prove that the chromatic symmetric function of any n-vertex tree containing a vertex of degree d ≥ log 2 n + 1 is not e-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any n-vertex connected graph containing a cut vertex whose deletion disconnects the graph into d ≥ log 2 n + 1 connected components is not e-positive. Furthermore we prove that any n-vertex bipartite graph, including all trees, containing a vertex of degree greater than ⌈ n 2 ⌉ is not Schur-positive, namely not a positive linear combination of Schur functions. In complete generality, we prove that if an n-vertex connected graph has no perfect matching (if n is even) or no almost perfect matching (if n is odd), then it is not e-positive. We hence deduce that many graphs containing the claw are not e-positive.

A set partition σ of [n] = {1, . . . , n} contains another set partition π if restricting σ to some S ⊆ [n] and then standardizing the result gives π. Otherwise we say σ avoids π. For all sets of patterns consisting of partitions of [3], the sizes of the avoidance classes were determined by Sagan and by Goyt. Set partitions are in bijection with restricted growth functions (RGFs) for which Wachs and White defined four fundamental statistics. We consider the distributions of these statistics over various avoidance classes, thus obtaining multivariate analogues of the previously cited cardinality results. This is the first in-depth study of such distributions. We end with a list of open problems.

In Stanley’s seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not e -positive, that is, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not e -positive. Moreover, one such family is additionally claw-free, thus establishing that the e -positivity of chromatic symmetric functions is in general not dependent on the existence of an induced claw or of a contraction to a claw.

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