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Motivated by Stanleys $mathbf{(3+1)}$-free conjecture on chromatic symmetric functions, Foley, Ho`{a}ng and Merkel introduced the concept of strong $e$-positivity and conjectured that a graph is strongly $e$-positive if and only if it is (claw, net)-free. In order to study strongly $e$-positive graphs, they further introduced the twinning operation on a graph $G$ with respect to a vertex $v$, which adds a vertex $v$ to $G$ such that $v$ and $v$ are adjacent and any other vertex is adjacent to both of them or neither of them. Foley, Ho`{a}ng and Merkel conjectured that if $G$ is $e$-positive, then so is the resulting twin graph $G_v$ for any vertex $v$. Based on the theory of chromatic symmetric functions in non-commuting variables developed by Gebhard and Sagan, we establish the $e$-positivity of a class of graphs called tadpole graphs. By considering the twinning operation on a subclass of these graphs with respect to certain vertices we disprove the latter conjecture of Foley, Ho`{a}ng and Merkel. We further show that if $G$ is $e$-positive, the twin graph $G_v$ and more generally the clan graphs $G^{(k)}_v$ ($k ge 1$) may not even be $s$-positive, where $G^{(k)}_v$ is obtained from $G$ by applying $k$ twinning operations to $v$.
We prove that for any tree with a vertex of degree at least six, its chromatic symmetric function is not $e$-positive, that is, it cannot be written as a nonnegative linear combination of elementary symmetric functions. This makes significant progres
In a 2016 ArXiv posting F. Bergeron listed a variety of symmetric functions $G[X;q]$ with the property that $G[X;1+q]$ is $e$-positive. A large subvariety of his examples could be explained by the conjecture that the Dyck path LLT polynomials exhibit
In channel flows a step on the route to turbulence is the formation of streaks, often due to algebraic growth of disturbances. While a variation of viscosity in the gradient direction often plays a large role in laminar-turbulent transition in shear
Inspired by the recent work of Chen and Fu on the e-positivity of trivariate second-order Eulerian polynomials, we show the e-positivity of a family of multivariate k-th order Eulerian polynomials. A relationship between the coefficients of this e-po
Context. It is generally agreed that hydrogenation reactions dominate chemistry on grain surfaces in cold, dense molecular cores, saturating the molecules present in ice mantles. Aims. We present a study of the low temperature reactivity of solid pha