Related papers: Down-up algebras and chromatic symmetric functions
We generalize Stembridge's enriched $P$-partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group. Whereas Stembridge's enriched $P$-partitions are related to…
Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\mathbb F_p$. Given a homogeneous equation $\mathcal L:\sum_{i=1}^k c_i x_i=0$ with $k\ge 3$, we study…
We expand the chromatic symmetric functions for Dyck paths of bounce number three in the elementary symmetric function basis using a combinatorial interpretation of the inverse of the Kostka matrix studied in E\u{g}ecio\u{g}lu-Remmel…
We establish a colorful and, more generally, matroidal solution to the problem of Goodman and Pollack on the existence of an $\mathbb{F}$-affine $k$-dimensional transversal to a family of convex sets in $\mathbb{F}^d$, where $0 \le k \le d…
The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras…
The characters of Kazhdan--Lusztig elements of the Hecke algebra over $S_n$ (and in particular, the chromatic symmetric function of indifference graphs) are completely encoded in the (intersection) cohomology of certain subvarieties of the…
We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid $H$, and an $H$-structure $h$ on a set $N$, there are proper colorings of $h$, generalizing graph…
We define a class of bipartite graphs that correspond naturally with Ferrers diagrams. We give expressions for the number of spanning trees, the number of Hamiltonian paths when applicable, the chromatic polynomial, and the chromatic…
We solve the isomorphism problem for non noetherian down-up algebras $A(\alpha,0,\gamma)$ by lifting isomorphisms between some of their non commutative quotients. The quotients we consider are either quantum polynomial algebras in two…
We explore the interplay between algebraic combinatorics and algorithmic problems in graph theory by defining a polynomial with connections to correspondence colouring (also known as DP-colouring), a recent generalization of list-colouring,…
In his work on P-partitions, Stembridge defined the algebra of peak functions Pi, which is both a subalgebra and a retraction of the algebra of quasi-symmetric functions. We show that Pi is closed under coproduct, and therefore a Hopf…
The so called generalized down-up algebras are revisited from a viewpoint of Gr\"obner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided $\lambda\omega\ne 0$), and…
The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$.…
In the present note we obtain new results on two conjectures by Csordas et al. regarding the interlacing property of zeros of special polynomials. These polynomials came from the Jacobi tau methods for the Sturm-Liouville eigenvalue…
Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $\pi$ angle of the skew…
The chromatic polynomial is a well studied object in graph theory. There are many results and conjectures about the log-concavity of the chromatic polynomial and other polynomials related to it. The location of the roots of these…
We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent…
We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko (GLM) equation. This is a linear integral equation in which…
The combinatorial Hopf algebra on building sets $BSet$ extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the…
We suggest a relatively simple and totally geometric conjectural description of uncolored DAHA superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov-Rozansky polynomials) via the flagged…