Published in Journal of Algebraic Combinatorics
We introduce the notion of a t-design on the Grassmann manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{G}_{m,n} $$ \end{document} of the m-subspaces ...
Published in Journal of Algebraic Combinatorics
For about 10 years, the classification up to Wilf equivalence of permutation patterns was thought completed up to length 6. In this paper, we establish a new class of Wilf-equivalent permutation patterns, namely, (n − 1, n − 2, n, τ) ∼ (n − 2, n, n − 1, τ) for any τ∈Sn−3. In particular, at level n = 6, this result includes the only missing equivale...
Published in Journal of Algebraic Combinatorics
Given a finite Weyl group W with root system Φ, assign the weight α ∈ Φ to each covering pair in the Bruhat order related by the reflection corresponding to α. Extending this multiplicatively to chains, we prove that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula, and prove a similar result for a we...
Published in Journal of Algebraic Combinatorics
As pointed out in Arbarello and Cornalba (J. Alg. Geom.5 (1996), 705–749), a theorem due to Di Francesco, Itzykson, and Zuber (see Di Francesco, Itzykson, and Zuber, Commun. Math. Phys.151 (1993), 193–219) should yield new relations among cohomology classes of the moduli space of pointed curves. The coefficients appearing in these new relations can...
Published in Journal of Algebraic Combinatorics
A t-cover of a quadric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Q$$ \end{document} is a set C of t-dimensional subspaces contained in \documentclass[12pt]{minim...
Published in Journal of Algebraic Combinatorics
We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type An − 1 by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in W coincide with the set of fully commutative elements;...
Published in Journal of Algebraic Combinatorics
Let Xn be either the symmetric group on n letters, the set of planar binary n-trees or the set of vertices of the (n − 1)-dimensional cube. In each case there exists a graded associative product on ⊕n≥0K[Xn]. We prove that it can be described explicitly by using the weak Bruhat order on Sn, the left-to-right order on planar trees, the lexicographic...
Published in Journal of Algebraic Combinatorics
Let Γ denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ0, θ1, ..., θD is a Q-polynomial ordering of the eigenvalues of Γ. This sequence is known to satisfy the recurrence θi − 1 − βθi + θi + 1 = 0 (0 > i > D), for some real scalar β. Let q denote a complex scalar such that q + q−1 = β. Bannai and Ito have co...
Published in Journal of Algebraic Combinatorics
If a GQ S′ of order (s, s) is contained in a GQ S of order (s, s2) as a subquadrangle, then for each point X of S\S′ the set of points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \be...
Published in Journal of Algebraic Combinatorics
Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain higher level Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices.
