On a refinement of the generalized Catalan numbers for Weyl groups

Abstract

Abstract. Let Φ be an irreducible crystallographic root system with Weyl group W , coroot latticeQ and Coxeter number h, spanning a Euclidean space V , and let m be a positive integer. It is known that the set of regions into which the fundamental chamber of W is dissected by the hyperplanes in V of the form (α, x) = k for α ∈ Φ and k = 1, 2, . . . , m is equinumerous to the set of orbits of the action of W on the quotientQ/ (mh + 1)Q. A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of Φ, are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case m = 1 and Φ = A n−1 .

Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs

“…This was first proved by Shi in terms of coroots [20,Theorem 5.2]. The statement here is formulated in terms of roots, which can be found in, e.g., [3,Lemma 2.4]. Surprisingly, there is no difference between compatible and strongly compatible sets in the case of type A.…”

Elliptic and q-Analogs of the Fibonomial Numbers

“…4 The q-Fibonacci analog of the rational Catalan numbers This number is equal to the number of lattice paths from (0, 0) to (m, n) that stay weakly above the main diagonal of the m × n rectangle. The study of these numbers (also in the non-coprime case), and their q-analog and q, t-analog generalizations, is connected to numerous relevant topics including rectangular diagonal harmonics and Macdonald polynomials [3,6,7,8,22], Shi hyperplane arrangements and affine Weyl groups [2,16,33], affine Springer fibers [16,19], affine Hecke algebras [11], knot theory [17,18], and representation theory of Cherednik algebras [13,14,15]. The q-Fibonacci analog of the m, n-Catalan number is defined as…”

“…The only difference is that in[27] the tiles below the path touching the bottom of the rectangle are required to be vertical dominos, while here this condition is required for the tiles below the path touching the path itself. This modification is essential to make our combinatorial model work 2. The elliptic and q-analogs of the Fibonacci numbers we use are different from the analogs considered, e.g., in[32].…”

Noncrossing Partitions in Surprising Locations

“…Arrange the transpositions from top to bottom in the order they are applied. In the example, the first transposition is (3,4) and the curves on the left are drawn so that highest crossing occurs between the strands that are third and fourth from the left. The next transposition is (2, 3) so the curves are drawn so that the next highest crossing occurs between the strands that are currently second and third from the left.…”