773 research outputs found
The Delta square conjecture
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 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 , reducing it to the same
case of the Delta conjecture, and the Schr\"{o}der case, i.e. the case . The latter provides a broad generalization of the
-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 and
. Finally, in the appendix, we give a new proof of the Delta conjecture at
.Comment: 27 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1807.0541
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
. 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.Comment: 21 pages, 3 figure
The Schr\"oder case of the generalized Delta conjecture
We prove the Schr\"oder case, i.e. the case , of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018)
for in terms of decorated partially
labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This
result extends the Schr\"oder case of the Delta conjecture proved in
(D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the
-Schr\"oder 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.Comment: 22 pages, 12 figure
The Brownian limit of separable permutations
We study random uniform permutations in an important class of
pattern-avoiding permutations: the separable permutations. We describe the
asymptotics of the number of occurrences of any fixed given pattern in such a
random permutation in terms of the Brownian excursion. In the recent
terminology of permutons, our work can be interpreted as the convergence of
uniform random separable permutations towards a "Brownian separable permuton".Comment: 45 pages, 14 figures, incorporating referee's suggestion
Area-width scaling in generalised Motzkin paths
13 pages, 3 figures19 pages, 3 figure
Theta operators, refined Delta conjectures, and coinvariants
We introduce the family of Theta operators indexed by symmetric
functions that allow us to conjecture a compositional refinement of the
Delta conjecture of Haglund, Remmel and Wilson for . We
show that the -variable Catalan theorem of Zabrocki is precisely the
Schr\"{o}der 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
, and prove the case , extending the
shuffle theorem of Carlsson and Mellit to a generalized shuffle theorem for
. Moreover we show how this implies the case of
our generalized Delta square conjecture for
, extending the
square theorem of Sergel to a generalized square theorem for
. 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 , leaving open the
problem of finding a dinv statistic that gives the whole symmetric function.Comment: 39 pages, 14 figure
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