• Corpus ID: 239009434

A horizontal-strip LLT polynomial is determined by its weighted graph

@inproceedings{Tom2021AHL,
  title={A horizontal-strip LLT polynomial is determined by its weighted graph},
  author={Foster Tom},
  year={2021}
}
We prove that two horizontal-strip LLT polynomials are equal if the associated weighted graphs defined by the author in a previous paper are isomorphic. This provides a sufficient condition for equality of horizontal-strip LLT polynomials and yields a welldefined LLT polynomial indexed by a weighted graph. We use this to prove some new relations between LLT polynomials and we explore a connection with extended chromatic symmetric functions. 

Figures from this paper

References

SHOWING 1-10 OF 39 REFERENCES
A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials
In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $G_\lambda(x;q)$ in some special cases. We associate a
LLT polynomials, chromatic quasisymmetric functions and graphs with cycles
Abstract We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some
A note on distinguishing trees with the chromatic symmetric function
  • Logan Crew
  • Computer Science, Mathematics
    Discret. Math.
  • 2022
TLDR
It is shown that the chromatic symmetric function of T determines the size of T ◦, and the multiset of the lengths of these incident paths, which generalizes a proof of Martin, Morin, and Wagner that the Chromatic symmetrical function distinguishes spiders.
The Roberts characterization of proper and unit interval graphs
  • F. Gardi
  • Computer Science, Mathematics
    Discret. Math.
  • 2007
TLDR
A constructive proof that the classes of proper interval graphs and unit interval graphs coincide is given, and a linear-time and space algorithm to compute a unit interval representation is yielded.
A deletion-contraction relation for the chromatic symmetric function
TLDR
This work extends the definition of the chromatic symmetric function to include graphs with a vertex-weight function w : V(G) with a natural deletion-contraction relation analogous to that of the Chromatic polynomial.
Proper caterpillars are distinguished by their chromatic symmetric function
We show that the symmetric function generalization of the chromatic polynomial, or equivalently, the U-polynomial, distinguishes among a large class of caterpillar trees that we call proper, thus
Chromatic quasisymmetric functions
We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's
Melting lollipop chromatic quasisymmetric functions and Schur expansion of unicellular LLT polynomials
TLDR
The chromatic quasisymmetric functions and the unicellular LLT polynomials are related via plethystic substitution and thus they satisfy the same linear relations, and the linear relations are applied to both sets of functions.
Chromatic quasisymmetric functions and Hessenberg varieties
We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics
A combinatorial formula for Macdonald polynomials
Abstract In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for Macdonald polynomials for all Lie types. These formulas resemble the formulas of Haglund,
...
1
2
3
4
...