arXiv subjects

David G. L. Wang

Publications and source records attributed to David G. L. Wang.

At least 19 recordsLinked to original sources

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.

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arXiv2026

Schur positivity from signed elementary expansions: clique-spiders and spiders $S(a,b,2)$

Determining when the chromatic symmetric function of a graph is Schur positive is a central problem in algebraic combinatorics. Elementary expansions can help answer this question, but negative coefficients obstruct a direct positivity argument. We develop methods for proving Schur positivity from signed elementary expansions and apply them to chromatic symmetric functions of clique-spiders and three-legged spiders. We first give the Schur alpha-omega lemma, which bounds Schur support. We then give four equivalent sufficient criteria for Schur positivity from a fixed composition-indexed elementary expansion. The matrix, order-ideal, negative Hall, and positive Hall formulations treat positive coefficients as capacities and negative coefficients as demands, with admissible assignments governed by dominance order. These criteria provide certificates of Schur positivity especially when elementary coefficients are negative. For the four basic families of $3$-clique-spiders with total leg length at most one, we obtain complete classifications of both $e$-positivity and Schur positivity. These classifications show how the position of a single leg edge changes positivity. Using selected clique-spider cases and a path-clique bootstrap, we prove that all spiders of the form $S(a,b,2)$ are Schur positive.

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arXiv2026

An $e$-positive classification for complete multipartite graphs

Shelburne and van Willigenburg (arXiv:2604.26158) characterize the Schur-positive complete multipartite graphs and leave open whether the graphs~$G=K_{(3,\,2^β)}$ are $e$-positive. We resolve this question and, together with their classification, characterize all $e$-positive complete multipartite graphs. Our main result is an explicit, manifestly nonnegative $e$-expansion of~$X_G$ whose coefficients are expressed in terms of the restricted-injection numbers. Our main idea is to derive a marker-variable coefficient-extraction formula for the $e$-coefficients of arbitrary complete multipartite graphs from the elementary--monomial Cauchy identity. For the particular graph~$G$, this formula reduces the proof to three coefficient families, which we evaluate using Dickson polynomials and recurrences for these numbers.

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arXiv2026

Schur positivity of three-legged spiders

We give a complete classification of Schur positivity for three-legged spiders. Every spider with at least two even legs is Schur positive, whereas every spider with three odd legs has a negative Schur coefficient. If $a$ is even and $b,c$ are odd, then the spider $S(a,b,c)$ is Schur positive if and only if $a\le 5b+5c+2$. Whenever Schur positivity fails, there is exactly one negative Schur coefficient, whose value we determine explicitly. These results extend those of Thibon and Wang for the families $S(a,2,1)$ and $S(a,4,1)$, and those of Wang and Wang for $S(a,b,2)$. The proof combines known Schur-positivity results for clique-spiders with Pieri's rule and simultaneous induction.

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arXiv2026

Schur positivity of the spiders $S(a,2,1)$ and $S(a,4,1)$ via noncommutative symmetric functions

We prove that the spider graphs $S(a,2,1)$ and $S(a,4,1)$ are Schur positive for all integers $a\ge1$. Together with the known $e$-positivity results, this completes the $e$- and Schur-positivity classification of both families. Our approach uses noncommutative symmetric functions, including a particularly simple ribbon expansion for the path lift with coefficients given by powers of two. We give a new proof of the Shareshian--Wachs path formula at $t=1$ and construct corresponding lifts for spiders. The Littlewood--Richardson rule converts their ribbon expansions into a general Schur-coefficient formula in terms of weighted Yamanouchi words. Mass-preserving multi-injections and a reduction to finitely many inequalities in degree $10$ then prove the required positivity; exact computer verification of these inequalities completes the proof in full generality.

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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.

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arXiv2026

Two infinite families of counterexamples to the Stanley--Gasharov conjecture

The Stanley--Gasharov conjecture asserts that every claw-free graph is Schur-positive. Prajapati and, independently, Matherne and Morales identified the same pair of counterexamples, both of which are line graphs, thereby disproving the conjecture. In this paper, we construct two infinite families of counterexamples to the Stanley--Gasharov conjecture, thereby answering a question of Matherne and Morales. Every graph in the first family is a line graph, whereas no graph in the second family is a line graph. Prajapati further showed that the graph $G_2$, which has $12$ vertices and $21$ edges, is the smallest counterexample under the ordering that first compares the numbers of vertices and then the numbers of edges. We show that $G_2$ is also the smallest counterexample under the reverse ordering, which first compares the edge numbers and then the vertex numbers. Similarly, we exhibit a graph $Q$ with $13$ vertices and $27$ edges and show that $Q$ is the smallest counterexample that is not a line graph under each ordering. Our two infinite families are obtained from $G_2$ and $Q$, respectively, by adjoining a clique of order at least $4$ and connecting one of its vertices to a distinguished vertex of the original graph by a single edge.

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arXiv2026

Exact thresholds for Schur positivity of the lattices $\mathbf m\times\mathbf 2$ and $\mathbf m\times\mathbf 3$

We determine the exact thresholds for Schur positivity in the two families of chain products $\mathbf m\times\mathbf2$ and $\mathbf m\times\mathbf3$: the former is Schur positive exactly for $m\le7$, and the latter exactly for $m\le6$. For $m\ge8$, we prove non-Schur-positivity in both families by exhibiting explicit negative Schur coefficients obtained from Pieri's rules and stable-composition counts. The remaining finite cases are settled by exact SageMath computations; in particular, $\mathbf7\times\mathbf3$ has a negative Schur coefficient. These results settle the $n=2$ and $n=3$ cases in the conjectural picture of Li, Qiu, Yang, and Zhang and sharpen the $n=3$ boundary by one. We also show that $\mathbf m\times\mathbf3$ is not strongly nice for $m\ge44$.

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arXiv2025

The trinacria graphs $T_{(b+2)b2}$ are $e$-positive

In this paper, we identify a new family of $e$-positive graphs, called the trinacria graphs $T_{(b+2)b2}$, thereby providing a partial answer to Stanley's question on which graphs are $e$-positive. The trinacria graph $T_{abc}$ is the graph on $a+b+c+3$ vertices obtained by attaching paths $P_a$, $P_b$ and~$P_c$ to the vertices of a triangle, respectively. Our proof relies on several ad hoc combinatorial ideas, and employs divide-and-conquer techniques, charging arguments, and progressive repair methods.

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arXiv2025

Clocks are $e$-positive

Along with his confirmation of the $e$-positivity of all cycle-chord graphs $θ_{ab1}$, the third author conjectured the $e$-positivity of all theta graphs $θ_{abc}$. In this paper, we establish the $e$-positivity of all clock graphs $θ_{ab2}$ by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least $2$.

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arXiv2025

All cycle-chords are $e$-positive

We establish the $e$-positivity of cycle-chord graphs by using the composition method which is developed by Zhou and the author recently. Our method is simpler than the $(e)$-positivity approach which is used for handling cycle-chords with girth at most $4$. We also provide a combinatorial interpretation of the $e$-coefficients, and conjecture that theta graphs are $e$-positive.

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arXiv2024

A composition method for neat formulas of chromatic symmetric functions

We develop a composition method to unearth positive $e_I$-expansions of chromatic symmetric functions $X_G$, where the subscript $I$ stands for compositions rather than integer partitions. Using this method, we derive positive and neat $e_I$-expansions for the chromatic symmetric functions of tadpoles, barbells and generalized bulls, and establish the $e$-positivity of hats. We also obtain a compact ribbon Schur analog for the chromatic symmetric function of cycles.

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arXiv2022

A unimodal sequence with mode at a quarter length

We show that the number $A(n,m)$ of partitions with $m$ even parts and largest hook length $n$ is strongly unimodal with mode [(n-1)/4] for $n\ge 6$. We establish this result by induction, using a $5$-term recurrence due to Lin, Xiong and Yan, and two $4$-term recurrences obtained by Zeilberger's algorithm. The sequence $A(n,m)$ is not log-concave. Using Möbius transformation and the method of interlacing zeros, we obtain that every zero of every generating function $\sum_m A(n,m)z^m$ lies on the left half part of the circle |z-1|=2. Moreover, as a direct application of Wang and Zhang's characterization of root geometry of polynomial sequences that satisfy a recurrence of type $(1,1)$, we see that all these zeros are densely distributed on the half circle.

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arXiv2021

The $e$-positivity and Schur positivity of the chromatic symmetric functions of some trees

We investigate the $e$-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods involve extracting particular $e$-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott's triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We also propose some conjectures on the $e$-positivity and Schur positivity of trees.

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arXiv2021

Two cycle-chord graphs are $e$-positive

We prove Gebhard and Sagan's $(e)$-positivity of the line graphs of tadpoles in noncommuting variables. This implies the $e$-positivity of these line graphs. We then extend this $(e)$-positivity result to that of certain cycle-chord graphs, and derive the bivariate generating function of all cycle-chord graphs.

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arXiv2021

A decomposition of ballot permutations, pattern avoidance and Gessel walks

A permutation whose any prefix has no more descents than ascents is called a ballot permutation. In this paper, we present a decomposition of ballot permutations that enables us to construct a bijection between ballot permutations and odd order permutations, which proves a set-valued extension of a conjecture due to Spiro using the statistic of peak values. This bijection also preserves the neighbors of the largest letter in permutations and thus resolves a refinement of Spiro' s conjecture proposed by Wang and Zhang. Our decomposition can be extended to well-labelled positive paths, a class of generalized ballot permutations arising from polytope theory, that were enumerated by Bernardi, Duplantier and Nadeau. We will also investigate the enumerative aspect of ballot permutations avoiding a single pattern of length 3 and establish a connection between 213-avoiding ballot permutations and Gessel walks.

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arXiv2020

Positivity and divisibility of alternating descent polynomials

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a quadratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{\lfloor n/2\rfloor}$ divides $\sum_{π\in\mathfrak{S}_n}q^{\rm{altmaj}(π)}$, where $\mathfrak{S}_n$ is the set of all permutations of $\{1,2,\ldots,n\}$ and $\rm{altmaj}(π)$ is the alternating major index of $π$. This leads us to discover a $q$-analog of $n!=2^{\ell}m$, $m$ odd, using the statistic of alternating major index. Moreover, we study the $γ$-vectors of the alternating descent polynomials by using these two recursions and the ${\textbf{cd}}$-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.

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arXiv2020

The twinning operation on graphs does not always preserve $e$-positivity

Motivated by Stanley's $\mathbf{(3+1)}$-free conjecture on chromatic symmetric functions, Foley, Hoàng 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, Hoàng 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, Hoàng 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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