## 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 → ∞.

Original language | English |
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Pages (from-to) | 1445-1460 |

Number of pages | 16 |

Journal | European Journal of Combinatorics |

Volume | 31 |

Issue number | 5 |

DOIs | |

State | Published - Jul 2010 |

### Bibliographical note

Funding Information:The authors would like to thank anonymous referees for helpful suggestions and comments. The second author was supported by the National Natural Science Foundation of China (No. 10901141).

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics