arXiv subjects

Alessandro Iraci

Publications and source records attributed to Alessandro Iraci.

26 records · Page 2Linked to original sources

arXiv2020

A valley version of the Delta square conjecture

Inspired by [Qiu, Wilson 2019] and [D'Adderio, Iraci, Vanden Wyngaerd 2019 - Delta Square], we formulate a generalised Delta square conjecture (valley version). Furthermore, we use similar techniques as in [Haglund, Sergel 2019] to obtain a schedule formula for the combinatorics of our conjecture. We then use this formula to prove that the (generalised) valley version of the Delta conjecture implies our (generalised) valley version of the Delta square conjecture. This implication broadens the argument in [Sergel 2016], relying on the formulation of the touching version in terms of the $Θ_f$ operators introduced in [D'Adderio, Iraci, Vanden Wyngaerd 2019 - Theta Operators].

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arXiv2019

Theta operators, refined Delta conjectures, and coinvariants

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 Wilson for $Δ_{e_{n-k-1}}'e_n$. We show that the $4$-variable Catalan theorem of Zabrocki is precisely the Schröder case of our compositional Delta conjecture, and we show how to relate this conjecture to the Dyck path algebra introduced by Carlsson and Mellit, extending one of their results. Again using the Theta operators, we conjecture a touching refinement of the generalized Delta conjecture for $Δ_{h_m}Δ_{e_{n-k-1}}'e_n$, and prove the case $k=0$, extending the shuffle theorem of Carlsson and Mellit to a generalized shuffle theorem for $Δ_{h_m}\nabla e_n$. Moreover we show how this implies the case $k=0$ of our generalized Delta square conjecture for $\frac{[n-k]_t}{[n]_t}Δ_{h_m}Δ_{e_{n-k}}ω(p_n)$, extending the square theorem of Sergel to a generalized square theorem for $Δ_{h_m}\nabla ω(p_n)$. Still the Theta operators will provide a conjectural formula for the Frobenius characteristic of super-diagonal coinvariants with two sets of Grassmanian variables, extending the one of Zabrocki for the case with one set of such variables. We propose a combinatorial interpretation of this last formula at $q=1$, leaving open the problem of finding a dinv statistic that gives the whole symmetric function.

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arXiv2019

The new dinv is not so new

In (Duane, Garsia, Zabrocki 2013) the authors introduced a new dinv statistic, denoted ndinv, on the two part case of the shuffle conjecture (Haglund et al. 2005) in order to prove a compositional refinement. Though in (Hicks, Kim 2013) a non-recursive (but algorithmic) definition of ndinv has been given, this statistic still looks a bit unnatural. In this paper we "unveil the mystery" around the ndinv, by showing bijectively that the ndinv actually matches the usual dinv statistic in a special case of the generalized Delta conjecture in (Haglund, Remmel, Wilson 2018). Moreover, we give also a non-compositional proof of the "$ehh$" case of the shuffle conjecture (after (Garsia, Xin, Zabrocki 2014)) by bijectively proving a relation with the two part case of the Delta conjecture.

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arXiv2019

The generalized Delta conjecture at t=0

We prove the cases q=0 and t=0 of the generalized Delta conjecture of Haglund, Remmel and Wilson involving the symmetric function $Δ_{h_m}Δ_{e_{n-k-1}}'e_n$. Our theorem generalizes recent results by Garsia, Haglund, Remmel and Yoo. This proves also the case q=0 of our recent generalized Delta square conjecture.

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arXiv2018

The Delta square conjecture

We conjecture a formula for the symmetric function $\frac{[n-k]_t}{[n]_t}Δ_{h_m}Δ_{e_{n-k}}ω(p_n)$ in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington (Loehr, Warrington 2007), recently proved by Sergel (Sergel 2017) after the breakthrough of Carlsson and Mellit (Carlsson, Mellit 2018). Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson (Haglund, Remmel, Wilson 2015), answering one of their questions. We support our conjecture by proving the specialization $m=q=0$, reducing it to the same case of the Delta conjecture, and the Schröder case, i.e. the case $\langle \cdot ,e_{n-d}h_d\rangle$. The latter provides a broad generalization of the $q,t$-square theorem of Can and Loehr (Can, Loehr 2006). We give also a combinatorial involution, which allows to establish a linear relation among our conjectures (as well as the generalized Delta conjectures) with fixed $m$ and $n$. Finally, in the appendix, we give a new proof of the Delta conjecture at $q=0$.

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arXiv2018

The Schröder case of the generalized Delta conjecture

We prove the Schröder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $Δ_{h_m}Δ_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This result extends the Schröder case of the Delta conjecture proved in (D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the $q,t$-Schröder of Haglund (Haglund 2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another combinatorial interpretation of the same polynomial in terms of a new bounce statistic. Moreover, we give two more interpretations of the same polynomial in terms of doubly decorated parallelogram polyominoes, extending some of the results in (D'Adderio, Iraci 2017), which in turn extended results in (Aval et al. 2014). Also, we provide combinatorial bijections explaining some of the equivalences among these interpretations.

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arXiv2017

Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture (FULL VERSION)

We introduce area, bounce and dinv statistics on decorated parallelogram polyominoes, and prove that some of their q,t-enumerators match $\langle Δ_{h_m} e_{n+1},s_{k+1,1^{n-k}}\rangle$, extending in this way the work in (Aval et al. 2014). Also, we provide a bijective connection between decorated parallelogram polyominoes and decorated labelled Dyck paths, which allows us to prove the combinatorial interpretation of the coefficient $\langle Δ_{e_{m+n-k-1}}'e_{m+n},h_m h_n\rangle$ predicted by the Delta conjecture in (Haglund et al. 2015). Finally, we define a statistic pmaj on partially labelled Dyck paths, which provides another conjectural combinatorial interpretation of $Δ_{h_{\ell}}Δ_{e_{n-k-1}}'e_n$, cf. (Haglund et al. 2015). This is the full version of (D'Adderio, Iraci 2017) arXiv:1711.03923.

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arXiv2017

Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture

We introduce area, bounce and dinv statistics on decorated parallelogram polyominoes, and prove that some of their q,t-enumerators match $\langle Δ_{h_m} e_{n+1}, s_{k+1,1^{n-k}} \rangle$, extending in this way the work in (Aval et al. 2014). Also, we provide a bijective connection between decorated parallelogram polyominoes and decorated labelled Dyck paths, which allows us to prove the combinatorial interpretation of the coefficient $\langleΔ_{e_{m+n-k-1}}'e_{m+n}, h_m h_n \rangle$ predicted by the Delta conjecture in (Haglund et al. 2015). Finally, we define a statistic pmaj on partially labelled Dyck paths, which provides another conjectural combinatorial interpretation of $Δ_{h_{\ell}}Δ_{e_{n-k-1}}'e_n$, cf. (Haglund et al. 2015). This is an extended abstract of (D'Adderio, Iraci 2017): this forthcoming publication will have proofs and additional details and results.

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