Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T11:10:28.737Z Has data issue: false hasContentIssue false
  • English
  • Français

A probabilistic interpretation of the Gaussian binomial coefficients

Part of: Stochastic processes Algebraic combinatorics

Published online by Cambridge University Press:  30 November 2017

Takis Konstantopoulos  and
Linglong Yuan
Takis Konstantopoulos*
Affiliation:
Uppsala University
Linglong Yuan*
Affiliation:
Xi'an Jiaotong-Liverpool University
*
* Postal address: Department of Mathematics, Uppsala University, SE-75106 Uppsala, Sweden. Email address: takiskonst@gmail.com
** Postal address: Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, 111 Ren'ai Road, Suzhou, 215123, P. R. China.
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a stand-alone simple proof of a probabilistic interpretation of the Gaussian binomial coefficients by conditioning a random walk to hit a given lattice point at a given time.

Keywords

Gaussian binomial coefficientq-binomialrandom walk

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 05E99: None of the above, but in this section
Type
Short Communications
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1295 - 1298
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Brent, R. P. and McKay, B. D. (1987). Determinants and ranks of random matrices over ℤ m . Discrete Math. 66, 35–49. Google Scholar
[2] Crippa, D. and Simon, K. (1995). q-distributions in random graphs: transitive closure and reduction. Tech. Rep., ETH Zürich. Google Scholar
[3] Crippa, D. and Simon, K. (1997). q-distributions and Markov processes. Discrete Math. 170, 81–98. CrossRefGoogle Scholar
[4] Gauss, C. F. (1811). Summatio quarundam serierum singularium. In Untersuchungen über höhere Arithmetik (translated by H. Maser), 2nd edn. Chelsea, New York, 1965. Google Scholar
[5] Kac, V. and Cheung, P. (2002). Quantum Calculus. Springer, New York. Google Scholar
[6] O'Connell, N. and Pei, Y. (2013). A q-weighted version of the Robinson–Schensted algorithm. Electron. J. Prob. 18, 95. CrossRefGoogle Scholar
[7] Pólya, G. (1969). On the number of certain lattice polygons. J. Combinatorial Theory 6, 102–105. CrossRefGoogle Scholar