Let R n be the ring of coinvariants for the diagonal action of the symmetric group S n . It is known that the character of R n as a doubly-graded S n module can be expressed using the Frobenius characteristic map as ∇e n , where e n is the n-th elementary symmetric function, and ∇ is an operator from the theory of Macdonald polynomials.We conjecture a combinatorial formula for ∇e n and prove that it has many desirable properties which support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc and Thibon. We also show that a variety of earlier conjectures and theorems on ∇e n are special cases of our conjecture.Finally, we extend our conjectures on ∇e n and several of the results supporting them to higher powers ∇ m e n .
International audience We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.
In a recent preprint, Carlsson and Oblomkov (2018) obtain a long sought after monomial basis for the ring DR n of diagonal coinvariants. Their basis is closely related to the "schedules" formula for the Hilbert series of DR n which was conjectured by the first author and Loehr (2005) and first proved by Carlsson and Mellit (2018), as a consequence of their proof of the famous Shuffle Conjecture. In this article we obtain a schedules formula for the combinatorial side of the Delta Conjecture, a conjecture introduced by the first author, Remmel and Wilson (2018) which contains the Shuffle Conjecture as a special case. Motivated by the Carlsson-Oblomkov basis for DR n and our Delta schedules formula, we introduce a (conjectural) basis for the module SDR n of super-diagonal coinvariants, an S n module generalizing DR n introduced recently by Zabrocki (2019) which conjecturally corresponds to the Delta Conjecture. arXiv:1908.04732v1 [math.CO]
In this note we will investigate a form of logic programming with constraints. The constraints that we consider will not be restricted to statements on real numbers as in CLP(R), see [15]. Instead our constraints will be arbitrary global constraints. The basic idea is that the applicability of a given rule is not predicated on the fact that individual variables satisfy certain constraints, but rather on the fact that the least model of the set rules that are ultimately applicable satisfy the constraint of the rule. Thus the role of clauses will be slightly different than in the usual Logic Programming with constraints. In fact, the paradigm we present is closely related to stable model semantics of general logic programming [13]. We will define the notion of a constraint model of our constraint logic program and show that stable models of logic programs as well as the supported models of logic programs are just special cases of constraint models of constraint logic programs.
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