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This paper is a study of the bases introduced by Chari–Loktev in [1] for local Weyl modules of the current algebra associated to a special linear Lie algebra. Partition overlaid patterns, POPs for short—whose introduction is one of the aims of this paper—form convenient parametrizing sets of these bases. They play a role analogous to that played by (Gelfand–Tsetlin) patterns in the representation theory of the special linear Lie algebra. The notion of a POP leads naturally to the notion of area of a pattern. We observe that there is a unique pattern of maximal area among all those with a given bounding sequence and given weight. We give a combinatorial proof of this and discuss its representation theoretic relevance. We then state a conjecture about the “stability”, i.e., compatibility in the long range, of Chari–Loktev bases with respect to inclusions of local Weyl modules. In order to state the conjecture, we establish a certain bijection between colored partitions and POPs, which may be of interest in itself. The stability conjecture has been proved in [6] in the rank one case. [ABSTRACT FROM AUTHOR]

Abstract: In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed examples of a distance graph on the real line R and difference graphs on the real plane R whose chromatic numbers depend upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct difference graphs on the real space R , whose chromatic number is a positive integer in the Zermelo–Fraenkel-choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. Ser. 2 (1970) 1). These examples illuminate how heavily combinatorial results can depend upon the underlying set theory, help appreciate the potential complexity of the chromatic number of n-space problem, and suggest that the chromatic number of n-space may depend upon the system of axioms chosen for set theory. [Copyright &y& Elsevier]

The polynomial method has been used recently to obtain many striking results in combinatorial geometry. In this paper, we use affine Hilbert functions to obtain an estimation theorem in finite field geometry. The most natural way to state the theorem is via a bounded degree Zariski closure operation: given a set, we consider all polynomials of some bounded degree vanishing on that set, and then the common zeros of these polynomials. For example, the degree d closure of d + 1 points on a line will contain the whole line, as any polynomial of degree at most d vanishing on the d + 1 points must vanish on the line. Our result is a bound on the size of a finite degree closure of a given set. Finally, we adapt our use of Hilbert functions to the method of multiplicities. [ABSTRACT FROM AUTHOR]

Abstract: The paper provides an upper bound on the size of a (generalized) separating hash family, a notion introduced by Stinson, Wei and Chen. The upper bound generalizes and unifies several previously known bounds which apply in special cases, namely bounds on perfect hash families, frameproof codes, secure frameproof codes and separating hash families of small type. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate subquadrangles of order s of flock generalized quadrangles of order , s odd, with base point , where the subquadrangle does not contain the point . We prove that if the flock generalized quadrangle has such a subquadrangle, then is classical. This solves “Remaining case (a)” in Brown and Thas [M.R. Brown, J.A. Thas, Subquadrangles of order s of generalized quadrangles of order , II, J. Combin. Theory Ser. A 106 (2004) 33–48] (“Remaining case (b)” was already handled in K. Thas [K. Thas, Symmetrieën in eindige veralgemeende vierhoeken (Symmetries in finite generalized quadrangles), Master thesis, Ghent University, Ghent, 1999, 186 p.]). As a corollary we have: if is an egg in for which the translation dual is good at the tangent space of at its element π and if there is a pseudo-oval on not containing π, then is classical. As a byproduct we prove that if the flock GQ of order , s odd, has a regular point x collinear with, but distinct from, the base point , then is a translation generalized quadrangle with base line . [Copyright &y& Elsevier]

Abstract: Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber''s formula for the sums of powers and a q-analogue of Gessel–Viennot''s formula involving Salié''s coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel–Viennot''s combinatorial interpretations. [Copyright &y& Elsevier]

Consider a set U of circles in the plane such that any line intersects at least one of those circles. For a given natural number m, is there a line in the plane intersecting at least m circles in U? In this paper this problem is solved. Our result is also generalized to compact convex subsets and to higher dimensional cases. [Copyright &y& Elsevier]

Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and Williams on the face enumeration of generalized permutohedra. They are γ -positive (in particular, palindromic and unimodal) polynomials which can be interpreted as h -polynomials of certain flag simplicial polytopes and which admit interesting Schur γ -positive symmetric function generalizations. This paper introduces analogues of these polynomials for r -colored permutations with similar properties and uncovers some new instances of equivariant γ -positivity in geometric combinatorics. [ABSTRACT FROM AUTHOR]