1. Bollobás set pair inequalities for compositionsAnyuan Tian, Yaokun Wu, 2025, original scientific article Abstract: A d-composition of a set S is an ordered d-tuple (S₁, …, S_d) where S₁, …, S_d are pairwise disjoint subsets of S. If we have a sequence of d-compositions of a finite set and observe certain intersection patterns among parts of different compositions, what are the corresponding arithmetic constraints on the parameters of this sequence? When d = 1, many results in extremal combinatorics address this question. Bollobás set pair inequality is such a classic result for d = 2. In this note, we provide several arithmetic constraints for general d and propose a conjecture as a linear space analogue for one of them. Our study highlights the connection between extremal combinatorics and Young’s lattice of a rectangle.
Keywords: Katona weight, Lubell weight, partition, shape homomorphism, Young's lattice of a rectangle
Published in RUP: 22.10.2025; Views: 232; Downloads: 5
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2. Ehrhart limitsBenjamin Braun, McCabe Olsen, 2025, original scientific article Abstract: We introduce the definition of an Ehrhart limit, that is, a formal power series with integer coefficients that is the limit in the ring of formal power series of a sequence of Ehrhart h*-polynomials. We identify a variety of examples of sequences of polytopes that yield Ehrhart limits, with a focus on reflexive polytopes and simplices.
Keywords: Ehrhart theory, lattice simplices, reflexive polytopes
Published in RUP: 16.09.2025; Views: 296; Downloads: 5
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3. Regular antilatticesKarin Cvetko-Vah, Michael Kinyon, Jonathan Leech, Tomaž Pisanski, 2019, original scientific article Abstract: Antilattices ▫$(S; \vee, \wedge)$▫ for which the Green's equivalences ▫$\mathcal{L}_{(\vee)}$▫, ▫$\mathcal{R}_{(\vee)}$▫, ▫$\mathcal{L}_{(\wedge)}$▫ and ▫$\mathcal{R}_{(\wedge)}$▫ are all congruences of the entire antilattice are studied and enumerated.
Keywords: noncommutative lattice, antilattice, Green's equivalences, lattice of subvarieties, enumeration, partition, composition
Published in RUP: 03.01.2022; Views: 2242; Downloads: 17
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7. Three-pencil lattice on triangulationsGašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 2007, published scientific conference contribution Abstract: In this paper, three-pencil lattices on triangulations are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to construct lattice points in a simple and numerically stable way. Further, this representation carries over to triangulations in a natural way. The construction is based upon group action of S 3 on triangle vertices, and it is shown that the number of degrees of freedom is equal to the number of vertices of the triangulation.
Keywords: numerical analysis, lattice, barycentric coordinates, triangulations, interpolation
Published in RUP: 03.04.2017; Views: 3325; Downloads: 92
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8. Barycentric coordinates for Lagrange interpolation over lattices on a simplexGašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 2008, published scientific conference contribution Abstract: In this paper, a ▫$(d+1)$▫-pencil lattice on a simplex in ▫${\mathbb{R}}^d$▫ is studied. The lattice points are explicitly given in barycentric coordinates. This enables the construction and the efficient evaluation of the Lagrange interpolating polynomial over a lattice on a simplex. Also, the barycentric representation, based on shape parameters, turns out to be appropriate for the lattice extension from a simplex to a simplicial partition.
Keywords: numerical analysis, lattice, barycentric coordinates, simplex, interpolation
Published in RUP: 03.04.2017; Views: 3505; Downloads: 146
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9. Lattices on simplicial partitionsGašper Jaklič, Jernej Kozak, Marjetka Knez, Vito Vitrih, Emil Žagar, 2010, published scientific conference contribution Abstract: In this paper, a ▫$(d+1)$▫-pencil lattices on a simplex in ▫${\mathbb{R}}^d$▫ are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of thisfact leads to an efficient computer algorithm for the design of a lattice.
Keywords: numerical analysis, lattice, barycentric coordinates, simplicial partition
Published in RUP: 03.04.2017; Views: 3344; Downloads: 139
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