Davion Q. B. Tang | Searcharxiv

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↗

arXiv2025

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

By using the composition method, we establish the $e$-positivity of spiders of the form $S(4m+2,\, 2m,\, 1)$, which was conjectured by Aliniaeifard, van Willigenburg and Wang. Following the divide-and-conquer strategy, we group one or two $e_J$-terms that have positive coefficients with each $e_I$-term that has a negative coefficient, where the compositions $J$ are selected to be obtained by rearranging the parts of $I$, and show the positivity of the sum of those coefficients. Our main contribution is an explicit construction of the injection.

math.CO↗