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Michele D'Adderio

Publications and source records attributed to Michele D'Adderio.

35 records · Page 2Linked to original sources
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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arXiv2017

Decorated Dyck paths, the Delta conjecture, and a new q,t-square

We discuss the combinatorics of the decorated Dyck paths appearing in the Delta conjecture framework of Haglund, Remmel and Wilson (2015) and Zabrocki (2016), by introducing two new statistics, bounce and bounce'. We then provide plethystic formulae for their q,t-enumerators, by proving in this way a decorated version of Haglund's q,t-Schroeder theorem, answering a question in Haglund, Remmel and Wilson (2015). In particular we provide both an algebraic and a combinatorial explanation of a symmetry conjectured in Haglund, Remmel and Wilson (2015) and Zabrocki (2016). Finally, we show a link between these results and the q,t-square theorem of Can and Loehr (2006) (conjectured by Loehr and Warrington (2007)), getting a new q,t-square, whose specialization at t=1/q is identical to the classical one, and hinting to a broader phenomenon of this type.

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arXiv2017

Brill-Noether theory of curves on $\mathbb{P}^1 \times \mathbb{P}^1$: tropical and classical approach

The gonality sequence $(d_r)_{r\geq1}$ of a smooth algebraic curve comprises the minimal degrees $d_r$ of linear systems of rank $r$. We explain two approaches to compute the gonality sequence of smooth curves in $\mathbb{P}^1 \times \mathbb{P}^1$: a tropical and a classical approach. The tropical approach uses the recently developed Brill--Noether theory on tropical curves and Baker's specialization of linear systems from curves to metric graphs. The classical one extends the work of Hartshorne on plane curves to curves on $\mathbb{P}^1 \times \mathbb{P}^1$.

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arXiv2016

The sandpile model on K_{m,n} and the rank of its configurations

We present an algorithm to compute the rank of a configuration of the sandpile model for the complete bipartite graph K_{m,n} of complexity O(m+n). Furthermore, we provide a formula for the generating function of parking sorted configurations on complete bipartite graphs K_{m,n} according to rank, degree, and the sizes m and n. The results in the present paper are similar to those found by Robert Cori and the second named author for the complete graph K_{n+1}, and they rely on the analysis of certain operators on the stable sorted configurations of K_{m,n} developed in a previous work by the authors together with Jean-Christophe Aval and Mark Dukes.

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arXiv2015

Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma

We introduce two operators on stable configurations of the sandpile model that provide an algorithmic bijection between recurrent and parking configurations. This bijection preserves their equivalence classes with respect to the sandpile group. The study of these operators in the special case of the complete bipartite graph ${K}_{m,n}$ naturally leads to a generalization of the well known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodic bi-infinite paths in the plane having slightly different slopes. We achieve our results by interpreting the action of these operators as an action on a point in the grid $\mathbb{Z}^2$ which is pointed to by one of these pairs of paths. Our Cyclic lemma allows us to enumerate several classes of polyominoes, and therefore builds on the work of Irving and Rattan (2009), Chapman et al. (2009), and Bonin et al. (2003).

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arXiv2013

Perfect powers in Catalan and Narayana numbers

When a Catalan number or a Narayana number is a (non-trivial) perfect power? For Catalan numbers, we show that the answer is "never". However, we prove that for every b, the Narayana number N(a,b) is a (non-trivial) perfect square for infinitely many values of a, and we show how to compute all of them. We also conjecture that N(a,b) is never a (non-trivial) perfect k-th power for k greater than 2 and we prove some cases of this conjecture.

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arXiv2013

Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [arXiv:1208.0024]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.

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arXiv2011

Graph colorings, flows and arithmetic Tutte polynomial

We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte.

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arXiv2011

Arithmetic matroids, Tutte polynomial, and toric arrangements

We introduce the notion of an arithmetic matroid, whose main example is given by a list of elements of a finitely generated abelian group. In particular we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula for the classical Tutte polynomial.

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arXiv2010

On a conjecture of Hivert and Thiéry about Steenrod operators

We prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics. In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture.

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arXiv2009

On isoperimetric profiles of algebras

Isoperimetric profile in algebras was first introduced by Gromov. We study the behavior of the isoperimetric profile under various ring theoretic constructions and its relation with the Gelfand-Kirillov dimension.

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