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Search: MSC category 05E99 ( None of the above, but in this section )

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1. CMB 2011 (vol 54 pp. 217)

Chen, William Y. C.; Wang, Larry X. W.; Yang, Arthur L. B.
Recurrence Relations for Strongly $q$-Log-Convex Polynomials
We consider a class of strongly $q$-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly $q$-log-convex. We also prove that the Bessel transformation preserves log-convexity.

Keywords:log-concavity, $q$-log-convexity, strong $q$-log-convexity, Bell polynomials, Bessel polynomials, Ramanujan polynomials, Dowling polynomials
Categories:05A20, 05E99

2. CMB 2009 (vol 53 pp. 3)

Athanasiadis, Christos A.
A Combinatorial Reciprocity Theorem for Hyperplane Arrangements
Given a nonnegative integer $m$ and a finite collection $\mathcal A$ of linear forms on $\mathcal Q^d$, the arrangement of affine hyperplanes in $\mathcal Q^d$ defined by the equations $\alpha(x) = k$ for $\alpha \in \mathcal A$ and integers $k \in [-m, m]$ is denoted by $\mathcal A^m$. It is proved that the coefficients of the characteristic polynomial of $\mathcal A^m$ are quasi-polynomials in $m$ and that they satisfy a simple combinatorial reciprocity law.

Categories:52C35, 05E99

3. CMB 2009 (vol 53 pp. 171)

Thomas, Hugh; Yong, Alexander
Multiplicity-Free Schubert Calculus
Multiplicity-free algebraic geometry is the study of subvarieties $Y\subseteq X$ with the ``smallest invariants'' as witnessed by a multiplicity-free Chow ring decomposition of $[Y]\in A^{\star}(X)$ into a predetermined linear basis. This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.

Categories:14M15, 14M05, 05E99

4. CMB 2005 (vol 48 pp. 460)

Sommers, Eric N.
$B$-Stable Ideals in the Nilradical of a Borel Subalgebra
We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

Categories:20F55, 17B20, 05E99

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