Mathematics

In [Haglund, Remmel, Wilson 2018] the authors state two versions of the so called Delta conjecture, the rise version and the valley version. Of the former, they also give a more general statement in which zero labels are also allowed. In [Qiu, Wilson 2020], the corresponding generalisation of the valley version is also formulated. In [D'Adderio, Iraci, Vanden Wyngaerd 2020], the authors use a pushing algorithm to prove the generalised version of the shuffle theorem. An extension of that argument is used in [Iraci, Vanden Wyngaerd 2020] to formulate a valley version of the (generalised) Delta square conjecture, and to suggest a symmetric function identity later stated and proved in [D'Adderio, Romero 2020]. In this paper, we use the pushing algorithm together with the aforementioned symmetric function identity in order to prove that the valley version of the Delta conjecture implies the valley version of the generalised Delta conjecture, which means that they are actually equivalent. Combining this with the results in [Iraci, Vanden Wyngaerd 2020], we prove that the valley version of the Delta conjecture also implies the corresponding generalised Delta square conjecture.

An Euler tour in a hypergraph H is a closed walk that traverses each edge of H exactly once, and an Euler family is a family of closed walks that jointly traverse each edge of H exactly once. An ??-covering k -hypergraph, for 2?��?<k , is a k -uniform hypergraph in which every ??-subset of vertices lie together in at least one edge. In this paper we prove that every ??-covering k -hypergraph, for k?? , admits an Euler family.

The (poset) cube Ramsey number R( Q n , Q n ) is defined as the least~ m such that any 2-coloring of the m -dimensional cube Q m admits a monochromatic copy of Q n . The trivial lower bound R( Q n , Q n )??n was improved by Cox and Stolee, who showed R( Q n , Q n )??n+1 for 3?�n?? and n??3 using a probabilistic existence proof. In this paper, we provide an explicit construction that establishes R( Q n , Q n )??n+1 for all n?? .

Let a and b be positive integers such that a≤b and a≡b(mod2) . We say that G has all (a,b) -parity factors if G has an h -factor for every function h:V(G)→{a,a+2,…,b−2,b} with b|V(G)| even and h(v)≡b(mod2) for all v∈V(G) . In this paper, we prove that every graph G with n≥3(b+1)(a+b) vertices has all (a,b) -parity factors if δ(G)≥( b 2 −b)/a , and for any two nonadjacent vertices u,v∈V(G) , max{ d G (u), d G (v)}≥ bn a+b . Moreover, we show that this result is best possible in some sense.

In this paper, we describe and prove a generalization of both the classical Greene-Kleitman duality theorem for posets and the local version proved recently by Lewis-Lyu-Pylyavskyy-Sen in studying discrete solitons, using an approach more closely linked to the approach of the classical case.

An expander code is a binary linear code whose parity-check matrix is the bi-adjacency matrix of a bipartite expander graph. We provide a new formula for the minimum distance of such codes. We also provide a new proof of the result that 2(1?��?γn is a lower bound of the minimum distance of the expander code given by a (m,n,d,γ,1?��? expander bipartite graph.

We define a new partial order on SY T n , the set of all standard Young tableaux with n cells, by combining the chain order with the notion of horizontal strips. We prove various desirable properties of this new order.

When are all positions of a game numbers? We show that two properties are necessary and sufficient. These properties are consequences of that, in a number, it is not an advantage to be the first player. One of these properties implies the other. However, checking for one or the other, rather than just one, can often be accomplished by only looking at the positions on the `board'. If the stronger property holds for all positions, then the values are integers.

We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph (V,G,<) has few induced copies of a small ordered hypergraph (W,H,?? then there is a small modification G ??so that (V, G ??,<) has no induced copies of (W,H,?? . (Note that we do \emph{not} need to modify the ordering < .) We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of ? -algebras. We then show that ordered hypergraphs can be viewed as hypergraphs where we view the intervals as an additional notion of a ``very structured'' set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon.

We generalize the shuffle theorem and its (km,kn) version, as conjectured by Haglund et al. and Bergeron et al., and proven by Carlsson and Mellit, and Mellit, respectively. In our version the (km,kn) Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of G L l characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for non-symmetric Hall-Littlewood polynomials.