Annals of Combinatorics 4 (2000) 195-197


A Bijective Answer to a Question of Zvonkin

Richard Ehrenborg

Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
jrge@math.kth.se

Received March 17, 2000

AMS Subject Classification: 05A15, 60C05

Abstract. The purpose of this note is to give a bijective proof of the identity

                                            $$E\left[<br>
\prod_{1 \leq i < j \leq n} (X_j - X_i)^2<br>
\right]<br>
=<br>
0! \cdot 1! \cdots n!,$$
where $X_1, \ldots, X_n$ are independent identically distributed normal random variables with mean 0 and variance 1. The bijection is obtained by combining a bijection of Gessel and a bijection of Ehrenborg with the interpretation that the moments of the normal distribution count the number of matchings.

Keywords: normal zero-one random variables, Vandermonde product, tournaments, matchings


References

1.  R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly 107 (2000) 557–560.

2.  I. Gessel, Tournaments and Vandermonde’s determinant, J. Graph Theory 3 (1979) 305–307.

3.  A. Zvonkin, Matrix integrals and map enumeration: An accessible introduction, Math. Comput. Modelling 26 (1997) 281–304.


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