#Weighted-bond symmetric functions
R. S. G. D'León and M. L. Wachs associate two symmetric functions with the multiweighted bond posets of a graph [DW26]. The first refines the Möbius invariant of the bond lattice; the second refines the chromatic polynomial. The latter is different from R. Stanley’s chromatic symmetric function.
#Multiweighted bond posets
Let \(G=(V,E).\) Its bond lattice \(\Pi_G\) is the subposet of the partition lattice consisting of set partitions whose blocks induce connected subgraphs of \(G.\)
For \(r\in\setP,\) an \(r\)-weighted partition has the form
\[\boldsymbol{\pi} =\{B_1^{\nu_1},\dotsc,B_k^{\nu_k}\},\]
where \(\{B_1,\dotsc,B_k\}\) is a set partition and each \(\nu_i\) is a weak \(r\)-composition of \(\lvert B_i\rvert-1.\) Refinement is allowed to merge blocks; when several blocks are merged, the new weight must dominate the sum of their old weights. Equivalently, a cover merges two blocks and adds one standard basis vector to the sum of their weights.
The \(r\)-weighted bond poset \(\Pi_G^r\) is the induced subposet on weighted partitions whose underlying partition lies in \(\Pi_G.\) We write \(\Pi_G^\infty\) when the weights are weak compositions with arbitrarily many parts. If \(G\) has \(n\) vertices and \(k\) connected components, then \(\Pi_G^r\) is pure of length \(n-k.\)
#The Möbius symmetric function
For a maximal element \(\boldsymbol{\pi}\) of \(\Pi_G^\infty,\) let \(w(\boldsymbol{\pi})\) be the sum of its block weights. The Möbius symmetric function of \(G\) is
\[M_G(\xvec) \coloneqq \sum_{\boldsymbol{\pi}\in\operatorname{Max}(\Pi_G^\infty)} \mu_{\Pi_G^\infty}(\widehat{0},\boldsymbol{\pi}) \xvec^{w(\boldsymbol{\pi})}.\]
It is a homogeneous symmetric function of degree \(n-k.\) If \(G\) is connected, symmetry gives the equivalent monomial expansion
\[M_G(\xvec) = \sum_{\lambda\vdash n-1} \mu_{\Pi_G^\infty}(\widehat{0},V^\lambda) \monomial_\lambda(\xvec).\]
The specializations \(M_G(1,0,0,\dotsc)\) and \(M_G(1,t,0,\dotsc)\) are, respectively, the Möbius invariant of \(\Pi_G\) and the Möbius polynomial of the ordinary weighted bond poset.
Example
Let \(P_3\) be the path on three vertices and \(K_3\) the complete graph. The recurrence below, computed exactly in the related Rust example, gives
\[M_{P_3}=\monomial_2+3\monomial_{11} =\elementaryE_{11}+\elementaryE_2,\]
and
\[M_{K_3}=2\monomial_2+5\monomial_{11} =2\elementaryE_{11}+\elementaryE_2.\]
The construction is multiplicative over connected components. If \(G\) is connected and has more than one vertex, then
\[\begin{aligned} M_G &=-\sum_{\pi\in\Pi_G\setminus\{\widehat{1}\}} \completeH_{\lvert\pi\rvert-1} \prod_{B\in\pi}M_{G|B},\\ M_G &=-\sum_{\pi\in\Pi_G\setminus\{\widehat{0}\}} M_{G/\pi} \prod_{B\in\pi}\completeH_{\lvert B\rvert-1}. \end{aligned}\]
For the path \(P_n,\) this family recovers the parking function symmetric function:
\[(-1)^{n-1}M_{P_n}=\omega\parking_{n-1} =\sum_{\pi\in NC_{n-1}}\elementaryE_{\lambda(\pi)}.\]
For a chordal graph with \(n\) vertices and \(k\) connected components, \((-1)^{n-k}M_G\) is elementary-positive. The corresponding statement for arbitrary graphs remains open.
Conjecture (González D’León–Wachs).
For every graph \(G\) with \(n\) vertices and \(k\) connected components, \((-1)^{n-k}M_G\) is elementary-positive.
#A chromatic symmetric analogue
Define
\[\Psi_G(\xvec) \coloneqq \sum_{\boldsymbol{\pi}\in\Pi_G^\infty} \mu_{\Pi_G^\infty}(\widehat{0},\boldsymbol{\pi}) \xvec^{w(\boldsymbol{\pi})}.\]
Equivalently,
\[\Psi_G(\xvec)=\sum_{\pi\in\Pi_G}M_{G|_\pi}(\xvec),\]
where \(G|_\pi\) is the spanning subgraph whose connected components are the induced graphs \(G|B,\) for \(B\in\pi.\) Its highest homogeneous component is \(M_G,\) and its one-variable specialization reverses the chromatic polynomial:
\[\Psi_G(t,0,0,\dotsc)=t^n\chi_G(t^{-1}).\]
For the two three-vertex examples above,
\[\begin{aligned} \Psi_{P_3} &=1-2\elementaryE_1+\elementaryE_{11}+\elementaryE_2,\\ \Psi_{K_3} &=1-3\elementaryE_1+2\elementaryE_{11}+\elementaryE_2. \end{aligned}\]
In particular, their one-variable specializations are the reversals of \(\chi_{P_3}(t)=t(t-1)^2\) and \(\chi_{K_3}(t)=t(t-1)(t-2),\) respectively.
Write \(f|_d\) for the homogeneous component of degree \(d.\) The function \(f\) is alternating elementary-positive if \((-1)^d f|_d\) is elementary-positive for every \(d.\) It is Schur-log-concave if
\[(f|_d)^2-(f|_{d-1})(f|_{d+1})\]
is Schur-positive for every \(d.\) Chordal graphs give alternating elementary-positive \(\Psi_G\); both of the following extensions are open.
Conjecture (González D’León–Wachs).
For every graph \(G,\) the symmetric function \(\Psi_G\) is alternating elementary-positive and Schur-log-concave.
The Schur-log-concavity conjecture has been checked for connected graphs with at most seven vertices. For example, the degree-one inequality for \(P_3\) is
\[(-2\elementaryE_1)^2 -(\elementaryE_{11}+\elementaryE_2) =3\schurS_{11}+2\schurS_2.\]
Finally, the first two equal-variable specializations for the complete graph are
\[\Psi_{K_n}(t,0,\dotsc)=\prod_{i=1}^{n-1}(t-i), \qquad \Psi_{K_n}(t,t,0,\dotsc)=(t-n)^{n-1}.\]
Finding comparably simple formulas after setting \(r\gt{}2\) variables equal to \(t\) is another open problem.
Bibliography
- [DW26]Rafael S. González D’León and Michelle L. Wachs. Weighted bond posets and a new chromatic symmetric function. arXiv:2608.08692v1, 2026.
.bib
@article{DLeonWachs2026x, author = {Rafael S. Gonz{\'a}lez D'Le{\'o}n and Michelle L. Wachs}, title = {Weighted bond posets and a new chromatic symmetric function}, year = {2026}, eprint = {2608.08692v1}, url = {https://arxiv.org/abs/2608.08692v1}, journal = {arXiv e-prints} }