Consider factorizations into transpositions of an n-cycle in the symmetric group S n. To every such factorization we assign a mono-mial in variables w ij that retains the transpositions used, but forgets their order. Summing over all possible factorizations of n-cycles we obtain a polynomial that happens to admit a closed expression. From this expression we… (More)
Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.
The article contains a generalization of the classical Whit-ney formula for the number of double points of a plane curve. This formula is split into a series of equalities, and also extended to curves on a torus, to non-pointed curves, and to wave fronts. All the theorems are given geometric proofs employing logarithmic Gauss-type maps from suitable… (More)
We introduce a new class of admissible pairs of triangular sequences and prove a bijection between the set of admissible pairs of triangular sequences of length n and the set of parking functions of length n. For all u and v = 0, 1, 2, 3 and all n ≤ 7 we describe in terms of admissible pairs the dimensions of the bi-graded components h u,v of diagonal… (More)