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Register a new userThe twinning operation on graphs does not always preserve $e$-positivity
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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 $v$.
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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 progress towards a recent conjecture of Dahlberg, She, and van Willigenburg, who conjectured the result for all trees with a vertex of degree at least four. We also provide a series of conditions that can identify when the chromatic symmetric function of a spider, a tree consisting of multiple paths identified at an end, is not $e$-positive. These conditions also generalize to trees and graphs with cut vertices. Finally, by applying a result of Orellana and Scott, we provide a method to inductively calculate certain coefficients in the elementary symmetric function expansion of the chromatic symmetric function of a spider, leading to further $e$-positivity conditions for spiders.
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 the same phenomenon. In this paper we list the results of computer explorations which suggest that other examples exhibit the same phenomenon. We prove two of the resulting conjectures and propose algorithms that would prove several of our conjectures. In writing this paper we have learned that similar findings have been independently discovered by Per Alexandersson.
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 flows, we show that it has, surprisingly, little effect on the algebraic growth. Non-uniform viscosity therefore may not always work as a flow-control strategy for maintaining the flow as laminar.
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-positive expansion and second-order Eulerian numbers is established. Moreover, we present a grammatical proof of the fact that the joint distribution of the ascent, descent and j-plateau statistics over k-Stirling permutations are symmetric distribution. By using symmetric transformation of grammars, a symmetric expansion of trivariate Schett polynomial is also established.
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 phase isocyanic acid (HNCO) with hydrogen atoms, with the aim of elucidating its reaction network. Methods. Fourier transform infrared spectroscopy and mass spectrometry were employed to follow the evolution of pure HNCO ice during bombardment with H atoms. Both multilayer and monolayer regimes were investigated. Results. The hydrogenation of HNCO does not produce detectable amounts of formamide (NH2CHO) as the major product. Experiments using deuterium reveal that deuteration of solid HNCO occurs rapidly, probably via cyclic reaction paths regenerating HNCO. Chemical desorption during these reaction cycles leads to loss of HNCO from the surface. Conclusions. It is unlikely that significant quantities of NH2CHO form from HNCO. In dense regions, however, deuteration of HNCO will occur. HNCO and DNCO will be introduced into the gas phase, even at low temperatures, as a result of chemical desorption.
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