The Ramanujan Journal

, Volume 37, Issue 2, pp 269–289 | Cite as

The generalized lucky ticket problem, perfect matchings, and closure relations satisfied by the Chebyshev and \(q\)-Hermite polynomials

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Abstract

Consideration is given to an asymmetric ticket of length \(m+n\) in base \(\ell \). Such a ticket is said to be \((m,n)\)-lucky if the sum of the first \(m\) digits is equal to that of the last \(n\) digits. In other words, a \((m,n)\)-lucky ticket is a \(m+n\) digit number (in base \(\ell \)) of the form \(a_1a_2\cdots a_mb_1b_2\cdots b_n\) where \(a_i,b_j \in \left\{ 0,1,2,\dots ,\ell -1 \right\} \) and \(a_1 + a_2 + \cdots + a_m = b_1 + b_2 + \cdots + b_n\). Applying both analytical (contour integral) and combinatorial methods, we arrive at two representations for the number of \((m,n)\)-lucky tickets in base \(\ell \). Our results reduce to those in the literature, when \(m=n\) and \(\ell =10\). 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 \(q\)-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 \(q=0\) 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.

Keywords

Lucky tickets Perfect matching Chebyshev polynomials \(q\)-Hermite polynomials Contour integration methods 

Mathematics Subject Classification

Primary 05C70 33D45 Secondary 05A15 
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Notes

Acknowledgments

The authors thank M. E. H. Ismail for suggesting a possible connection between the lucky ticket and the perfect matching problem, as well as for many fruitful discussions.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA

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