Area of Catalan paths on a checkerboard

Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

“…{p, q} = {2k − 1, 2k}). In this example, the component Γ (1) and Γ (2) has 6 and 4 vertices, respectively, and hence Ξ 5 (σ ) = (3, 2) and ν 5 (σ ) = 2.…”

“…Example 2.1 For σ = 1 2 3 4 5 6 7 8 9 10 5 1 4 10 3 9 7 6 2 8 ∈ S 10 , its graph Γ (σ ) has two connected components Γ (1) and Γ (2) :…”

“…The reduced coset-types of m, n, and m ∪ n are (1), (1), and (1, 1), respectively. Lemma 6.6 Let n (1) , n (2) ∈ M(2n) be such that k < l for all k ∈ S(n (1) ) and l ∈ S(n (2) ). Suppose that the reduced coset-types of n (i) have sizes r i (i = 1, 2).…”

“…Also, we define λ ∪ μ to be the partition whose parts are those of λ and μ, arranged in decreasing order. In general, given partitions λ (1) , λ (2) , . .…”

“…Its definition is based on Jack polynomial theory, and the connections between them and random matrix theory are much studied, see [1,18] and their references. From symmetric function theory we can see that A μ (F, n) = A (1) μ (F, n) and A μ (F, n) = A (2) μ (F, n) for any symmetric function F and partition μ. Also A (1/2) μ (F, n) are important and related to a twisted Gelfand pair.…”