The generalized lucky ticket problem, perfect matchings, and closure relations satisfied by the Chebyshev and q-Hermite polynomials
Abbreviated Journal Title
Lucky tickets; Perfect matching; Chebyshev polynomials; q-Hermite; polynomials; Contour integration methods; LINEARIZATION COEFFICIENTS; COMBINATORICS; LAGUERRE; Mathematics
Consideration is given to an asymmetric ticket of length in base . Such a ticket is said to be -lucky if the sum of the first digits is equal to that of the last digits. In other words, a -lucky ticket is a digit number (in base ) of the form where and . Applying both analytical (contour integral) and combinatorial methods, we arrive at two representations for the number of -lucky tickets in base . Our results reduce to those in the literature, when and . Furthermore, through the contour integral approach, we arrive at a non-obvious closure relation satisfied by the Chebyshev polynomials. The weighted ticket problem is also considered, and analogous results are obtained. As addressed by Ismail, Stanton, and Viennot, the generating function of the crossing numbers over perfect matchings is related to closure relations of -Hermite polynomials. In the second part of this paper, we give corresponding contour integral representations for these closure relations, which permit us to give an alternate representation of the number of perfect matchings between sets. In the limit, we obtain a representation equivalent to that of De Sainte-Catherine and Viennot for the number of Dyck words of a fixed length satisfying a set of algebraic restrictions. In order to relate the two combinatorial problems, we find an explicit correspondence between our contour formulations for each problem.
"The generalized lucky ticket problem, perfect matchings, and closure relations satisfied by the Chebyshev and q-Hermite polynomials" (2015). Faculty Bibliography 2010s. 6436.