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- W2052838603 abstract "Recently, Bouvel and Pergola initiated the study of a special class of permutations, minimal permutations with a given number of descents, which arise from the whole genome duplication–random loss model of genome rearrangement. In this paper, we show that the number of minimal permutations of length 2 d − 1 with d descents is given by 2 d − 3 ( d − 1 ) c d , where c d is the d -th Catalan number. For fixed n , we also derive a recurrence relation on the multivariate generating function for the number of minimal permutations of length n counted by the number of descents, and the values of the first and second elements of the permutation. For fixed d , on the basis of this recurrence relation, we obtain a recurrence relation on the multivariate generating function for the number of minimal permutations of length n with n − d descents, counted by the length, and the values of the first and second elements of the permutation. As a consequence, the explicit generating functions for the numbers of minimal permutations of length n with n − d descents are obtained for d ≤ 5 . Furthermore, we show that for fixed d ≥ 1 , there exists a constant a d such that the number of minimal permutations of length n with n − d descents is asymptotically equivalent to a d d n , as n → ∞ ." @default.
- W2052838603 created "2016-06-24" @default.
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- W2052838603 date "2010-07-01" @default.
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- W2052838603 title "Minimal permutations with <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si10.gif display=inline overflow=scroll><mml:mi>d</mml:mi></mml:math> descents" @default.
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- W2052838603 doi "https://doi.org/10.1016/j.ejc.2010.01.001" @default.
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