This work contains detailed descriptions of developments in the combinatorics of the space
of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics …

After recalling the basic notions of combinatorics, representation theory, and commutative
algebra, the main material in this book provides links between the study of …

We introduce the family of Theta operators Θ f indexed by symmetric functions f that allow us
to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and …

We define a family of maps on lattice paths, called sweep maps, that assign levels to each
step in the path and sort steps according to their level. Surprisingly, although sweep maps …

The modified Macdonald polynomials, introduced by Garsia and Haiman (1996), have many
astounding combinatorial properties. One such class of properties involves applying the …

We discuss the combinatorics of decorated Dyck paths and decorated parallelogram
polyominoes, extending to the decorated case the main results of both Haglund (“A proof of …

In The Delta Conjecture (arxiv: 1509.07058), Haglund, Remmel and Wilson introduced a
four variable $ q, t, z, w $ Catalan polynomial, so named because the specialization of this …

We conjecture a combinatorial formula for the monomial expansion of the image of any
Schur function under the Bergeron–Garsia nabla operator. The formula involves nested …

We conjecture a formula for the symmetric function in terms of decorated partially labelled
square paths. This can be seen as a generalization of the square conjecture of Loehr and …

Abstract Inspired by Qiu and Wilson (J Combin Theory Ser A 175: 105271, 2020) and
D'Adderio et al.(Eur J Combin 81: 58–83, 2019), we formulate a generalised Delta square …