Cumulants, lattice paths, and orthogonal polynomials
نویسندگان
چکیده
منابع مشابه
O ct 2 00 1 CUMULANTS , LATTICE PATHS , AND ORTHOGONAL POLYNOMIALS
A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.
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The rth Faber polynomial of the Laurent series f(t) = t + f0 + f1/t + f2/t + · · · is the unique polynomial Fr(u) of degree r in u such that Fr(f) = tr + negative powers of t. We apply Faber polynomials, which were originally used to study univalent functions, to lattice path enumeration.
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In their fundamental paper [18] Kazhdan and Lusztig defined, for every Coxeter group W , a family of polynomials, indexed by pairs of elements of W , which have become known as the Kazhdan-Lusztig polynomials of W (see, e.g., [17], Chap. 7). These polynomials are intimately related to the Bruhat order of W and to the geometry of Schubert varieties, and have proven to be of fundamental importanc...
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Let d(n) count the lattice paths from (0, 0) to (n, n) using the steps (0,1), (1,0), and (1,1). Let e(n) count the lattice paths from (0, 0) to (n, n) with permitted steps from the step set N × N − {(0, 0)}, where N denotes the nonnegative integers. We give a bijective proof of the identity e(n) = 2n−1d(n) for n ≥ 1. In giving perspective for our proof, we consider bijections between sets of la...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2003
ISSN: 0012-365X
DOI: 10.1016/s0012-365x(02)00834-8