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Expected Number of Inversions After a Sequence of Random Adjacent Transpositions

In: Formal Power Series and Algebraic Combinatorics

Author

Listed:
  • Henrik Eriksson

    (NADA, KTH)

  • Kimmo Eriksson

    (IMa, Mälardalens högskola)

  • Jonas Sjöstrand

Abstract

We give expressions for the expected number of inversions after t random adjacent transpositions have been performed on the identity permutation in S n + 1 The problem is a simplification of a problem motivated by genome evolution. For a fixed t and for all n ≥ t, the expected number of inversions after t random adjacent transpositions is $${E_{nt}} = t - \frac{2}{n}(\begin{array}{*{20}{c}} t\\ 2 \end{array}) + r = \sum\limits_{r = 2}^t {\frac{{{{( - 1)}^r}}}{{{n^r}}}} [{2^r}{C_r}(\begin {array}{*{20}{c}} t\\ {r + 1} \end{array}) + 4{d_r}(\begin{array}{*{20}{c}} t \\ r\end{array})],$$ where d 2 = 0, d 3 = 1, d 4 = 9, d 5 = 69,... is a certain integer sequence. An important part of the our method is the use of a heat conduction analogy of the random walks, which guarantees certain properties of the solution.

Suggested Citation

  • Henrik Eriksson & Kimmo Eriksson & Jonas Sjöstrand, 2000. "Expected Number of Inversions After a Sequence of Random Adjacent Transpositions," Springer Books, in: Daniel Krob & Alexander A. Mikhalev & Alexander V. Mikhalev (ed.), Formal Power Series and Algebraic Combinatorics, pages 677-685, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-04166-6_66
    DOI: 10.1007/978-3-662-04166-6_66
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