### A bijective proof of the hook-content formula for super Schur functions and a modified jeu de taquin.

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We study the cohomology ring of the Grassmannian G of isotropic n-subspaces of a complex 2m-dimensional vector space, endowed with a nondegenerate orthogonal form (here 1 ≤ n < m). We state and prove a formula giving the Schubert class decomposition of the cohomology products in H*(G) of general Schubert classes by "special Schubert classes", i.e. the Chern classes of the dual of the tautological vector bundle of rank n on G. We discuss some related properties of reduced decompositions of "barred...

Let K be an algebraic number field with non-trivial class group G and ${}_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let ${F}_{k}\left(x\right)$ denote the number of non-zero principal ideals ${a}_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that ${F}_{k}\left(x\right)$ behaves, for x → ∞, asymptotically like $x{\left(logx\right)}^{1/\left|G\right|-1}{\left(loglogx\right)}^{{}_{k}\left(G\right)}$. In this article, it is proved that for every prime p, $\u2081\left({C}_{p}\oplus {C}_{p}\right)=2p$, and it is also proved that $\u2081\left({C}_{mp}\oplus {C}_{mp}\right)=2mp$ if $\u2081\left({C}_{m}\oplus {C}_{m}\right)=2m$ and m is large enough. In particular, it is shown that for...

We consider four combinatorial interpretations for the algebra of Boolean differential operators and construct, for each interpretation, a matrix representation for the algebra of Boolean differential operators.

For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$; the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$. Its maximal cones are naturally indexed by the $c$-sortable elements of $W$. The main result of this paper is that the known bijection cl${}_{c}$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for $W$. The rays...