Recent zbMATH articles in MSC 06A07https://www.zbmath.org/atom/cc/06A072022-05-16T20:40:13.078697ZWerkzeugCatalan recursion on externally ordered bases of unit interval positroidshttps://www.zbmath.org/1483.050172022-05-16T20:40:13.078697Z"Camacho, Jan Tracy"https://www.zbmath.org/authors/?q=ai:camacho.jan-tracy"Chavez, Anastasia"https://www.zbmath.org/authors/?q=ai:chavez.anastasiaSummary: The Catalan numbers form a sequence that counts over 200 combinatorial objects. A remarkable property of the Catalan numbers, which extends to these objects, is its recursive definition; that is, we can determine the \(n\)-th object from previous ones. A matroid is a combinatorial object that generalizes the notion of linear independence with connections to many fields of mathematics. A family of matroids, called unit interval positroids (UIP), are Catalan objects induced by the antiadjacency matrices of unit interval orders. Associated to each UIP is the set of externally ordered bases, which due to \textit{M. Las Vergnas} [Eur. J. Comb. 22, No. 5, 709--721 (2001; Zbl 0984.52018)], produces a lattice after adjoining a bottom element. We study the poset of externally ordered UIP bases and the implied Catalan-induced recursion. Explicitly, we describe an algorithm for constructing the lattice of a rank-\(n\) UIP from the lattice of lower ranks. Using their inherent combinatorial structure, we define a simple formula to enumerate the bases for a given UIP.The edge-product space of phylogenetic trees is not shellablehttps://www.zbmath.org/1483.050242022-05-16T20:40:13.078697Z"Stadnyk, Grace"https://www.zbmath.org/authors/?q=ai:stadnyk.graceSummary: The edge-product space of phylogenetic trees is a regular CW complex whose maximal closed cells correspond to trivalent trees with leaves labeled by a finite set \(X\). The face poset of this cell decomposition is isomorphic to the Tuffley poset, a poset of labeled forests, with a unique minimum adjoined. We show that the edge-product space of phylogenetic trees is gallery-connected. We then use combinatorial properties of the Tuffley poset and a related graph known as NNI-tree space to show that, although open intervals of the Tuffley poset were proven to be shellable by \textit{J. Gill} et al. [ibid. 41, No. 2, 158--176 (2008; Zbl 1149.05305)], the edge-product space is not shellable.Duality of graded graphs through operadshttps://www.zbmath.org/1483.180232022-05-16T20:40:13.078697Z"Giraudo, Samuele"https://www.zbmath.org/authors/?q=ai:giraudo.samueleGiven a graded set \(G\), a graded graph on this set is a graph whose set of vertices is \(G\) and such that the grading difference between the ends of an edge is exactly one. It is possible to associate to such graphs generating series which count paths on them. For a certain notion of duality on these graphs, due to Fomin, one can recover some combinatorial identities. For instance, the Cauchy identity relating the number of standard Young tableaux and the number of permutations is given by the paths on the Young lattice and its dual graph. The author uses non-symmetric operads, an algebraic structure encoding products, to construct pairs of such dual graded graphs, with adjacency relations given by composition with a generator of the operad. These graphs are called prefix graded graphs. The author study the properties of the associated poset and describes its intervals. He also generalizes Fomin's duality and applies this construction to several non-symmetric operads such as the associative and the diassociative operads, integer compositions, Motzkin paths and m-trees operads.
Reviewer: Bérénice Delcroix-Oger (Paris)Tukey order and diversity of free abelian topological groupshttps://www.zbmath.org/1483.220032022-05-16T20:40:13.078697Z"Gartside, Paul"https://www.zbmath.org/authors/?q=ai:gartside.paul-mSummary: For a Tychonoff space \(X\) the \textit{free abelian topological group} over \(X\), denoted \(A(X)\), is the free abelian group on the set \(X\) with the coarsest topology so that for any continuous map of \(X\) into an abelian topological group its canonical extension to a homomorphism on \(A(X)\) is continuous.
We show there is a family \(\mathcal{A}\) of maximal size, \(2^{\mathfrak{c}}\), consisting of separable metrizable spaces, such that if \(M\) and \(N\) are distinct members of \(\mathcal{A}\) then \(A(M)\) and \(A(N)\) are not topologically isomorphic (moreover, \(A(M)\) neither embeds topologically in \(A(N)\) nor is an open image of \(A(N))\). We show there is a chain \(\mathcal{C}=\{M_\alpha:\alpha<\mathfrak{c}^+\}\), of maximal size, of separable metrizable spaces such that if \(\beta < \alpha\) then \(A( M_\beta)\) embeds as a closed subgroup of \(A( M_\alpha)\) but no subspace of \(A( M_\beta)\) is homeomorphic to \(A( M_\alpha)\).
We show that the character (minimal size of a local base at 0) of \(A(M)\) is \(\mathfrak{d}\) (minimal size of a cofinal set in \(\mathbb{N}^{\mathbb{N}})\) for every non-discrete, analytic \(M\), but consistently there is a co-analytic \(M\) such that the character of \(A(M)\) is strictly above \(\mathfrak{d}\).
The main tool used for these results is the Tukey order on the neighborhood filter at 0 in an \(A(X)\), and a connection with the family of compact subsets of an auxiliary space.