Alessandro Iraci

In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a ( m + k ) × ( n + k ) $(m+k) \times (n+k)$ left parenthesis m plus k right parenthesis times left parenthesis n plus k right parenthesis rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and nondecorated rectangular objects. We prove a symmetric function formula for the q , t $q,t$ q comma t -generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to e m , n + k m $e_{m,n+km}$ e Subscript m comma n plus k m and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths.

In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a $(m+k) \times (n+k)$ rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and non-decorated rectangular objects. We prove a symmetric function formula for the $q,t$-generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to $e_{m,n+km}$ and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths.

We give an elementary symmetric function expansion for the expressions $M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$ and $M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$ when $t=1$ in terms of what we call $\gamma $ -parking functions and lattice $\gamma $ -parking functions. Here, $\Delta _F$ and $\Pi $ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$ . Our main results, in turn, give an elementary basis expansion at $t=1$ for symmetric functions of the form $M \Delta _{Fe_1} \Theta _{G} J$ wheneverFis expanded in terms of monomials,Gis expanded in terms of the elementary basis, andJis expanded in terms of the modified elementary basis $\{\Pi e_\lambda ^\ast \}_\lambda $ . Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving ane-positivity conjecture for whentis not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.

We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs $\widehat{G}_{μ,ν}$, which we call clique-independent graphs, indexed by two compositions $μ$ and $ν$. Moreover, we define a delay statistic on these configurations, and we show that, together with the usual level statistic, it can be used to provide a new combinatorial interpretation of the celebrated shuffle theorem of Carlsson and Mellit. More precisely, we will see how to interpret the polynomials $\langle \nabla e_n, e_μh_ν\rangle$ in terms of these configurations.

We provide a combinatorial interpretation of the symmetric function $\left.Θ_{e_k}Θ_{e_l}\nabla e_{n-k-l}\right|_{t=0}$ in terms of segmented Smirnov words. The motivation for this work is the study of a diagonal coinvariant ring with one set of commuting and two sets of anti-commuting variables, whose Frobenius characteristic is conjectured to be the symmetric function in question. Furthermore, this function is related to the Delta conjectures. Our work is a step towards a unified formulation of the two versions, as we prove a unified Delta theorem at $t=0$.

We provide a combinatorial interpretation of the symmetric function $\Theta_{e_k}\Theta_{e_l}\nabla e_{n-k-l}\vert_{t=0}$ in terms of Smirnov words, which are words where adjacent letters are distinct. The motivation for this work is the study of a diagonal coinvariant ring with one set of commuting and two sets of anti-commuting variables, whose Frobenius characteristic is conjectured to be the symmetric function in question. It is intimately related to the two Delta conjectures, as our work is a step towards a unified formulation of these.

In this paper, we extend the rectangular side of the shuffle conjecture by stating a rectangular analogue of the square paths conjecture. In addition, we describe a set of combinatorial objects and one statistic that are a first step towards a rectangular extension of (the rise version of) the Delta conjecture, and of (the rise version of) the Delta square conjecture, corresponding to the case \(q=1\) of an expected general statement. We also prove our new rectangular paths conjecture in the special case when the sides of the rectangle are coprime.Mathematics Subject Classifications: 05E05Keywords: Macdonald polynomials, symmetric functions

Haglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018) introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function $$\Delta '_{e_{n-k-1}} e_n$$ Δ e n - k - 1 ′ e n in terms of rise-decorated or valley-decorated labelled Dyck paths. While the rise version has been recently proved (D’Adderio and Mellit in Adv Math 402:108342, 2022; Blasiak et al. in A Proof of the Extended Delta Conjecture, arXiv:2102.08815, 2021), not much is known about the valley version. In this work, we prove the Schröder case of the valley Delta conjecture, the Schröder case of its square version (Iraci and Wyngaerd in Ann Combin 25(1):195–227, 2021), and the Catalan case of its extended version (Qiu and Wilson in J Combin Theory Ser A 175:105271, 2020). Furthermore, assuming the symmetry of (a refinement of) the combinatorial side of the extended valley Delta conjecture, we deduce also the Catalan case of its square version (Iraci and Wyngaerd 2021).