Increasing Subsequences and the Classical Groups

Creators: Rains, E. M.

Abstract

We show that the moments of the trace of a random unitary matrix have combinatorial interpretations in terms of longest increasing subsequences of permutations. To be precise, we show that the 2n-th moment of the trace of a random k-dimensional unitary matrix is equal to the number of permutations of length n with no increasing subsequence of length greater than k. We then generalize this to other expectations over the unitary group, as well as expectations over the orthogonal and symplectic groups. In each case, the expectations count objects with restricted "increasing subsequence" length.

Additional Information

© 1998 Electronic Journal of Combinatorics. Submitted: November 24, 1997; Accepted: January 30, 1998. The greatest thanks are due to P. Diaconis (the author's thesis advisor), for his many suggestions of problems (one of which appears in the present work). Thanks are also due to the following people and institutions, in no particular order: A. Odlyzko, for many helpful comments; AT&T Bell Laboratories (Murray Hill) and the Center for Communications Research (Princeton) for generous summer support; the Harvard University Mathematics Department and the National Science Foundation, for generous support for the rest of the year. Also, thanks are due, for helpful comments, to N. Bergeron, M. Grassl, M. Rojas, R. Stanley, and D. Stroock. Last, but not least, the author owes thanks to W. Woyczynski, both for introducing him to probability theory and for introducing him to Prof. Diaconis; the thesis of which this work is a part would be very different, had either introduction not been made.

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