Davion Q. B. Tang | Search arXiv

arXiv2026

The spiders $S(4m+2,\,2m,\,1)$ are $e$-positive

Using the composition method, we establish the $e$-positivity of spiders of the form $S(4m+2,\, 2m,\, 1)$, as conjectured by Aliniaeifard, van Willigenburg and Wang. Following a divide-and-conquer strategy, we group one or two positive-coefficient $e_J$-terms with each negative-coefficient $e_I$-term, where each $J$ is a composition obtained by rearranging the parts of the composition $I$, and prove that the resulting total coefficient is nonnegative. Our main contribution is the explicit construction of the required injections.

math.CO↗

arXiv2026

Positive $e_I$-expansions for KPKP graphs, twinned lollipops, and kayak paddles

We derive explicit positive $e_I$-expansions for the chromatic symmetric functions of several graph families. First, we obtain a uniform formula for KPKP graphs that specializes to known formulas for lollipops, KPK graphs, KKP graphs, and PKP graphs. We then give positive $e_I$-expansions for twinned paths and cycles and apply the KPKP formula to twinned lollipops. Finally, we establish a positive $e_I$-expansion for kayak paddles and specialize it to infinity graphs. These formulas refine ordinary $e$-positivity by retaining composition-indexed coefficients and provide independent proofs complementary to existing recurrence, unit-interval, and noncommutative methods.

math.CO↗