On abab-free and abba-free set partitions
European Journal of Combinatorics
Polygon dissections and standard Young tableaux
Journal of Combinatorial Theory Series A
Discrete Mathematics - Special issue on selected papers in honor of Henry W. Gould
Generating functions for generating trees
Discrete Mathematics
Sorting by Transpositions Based on the First Increasing Substring Concept
BIBE '04 Proceedings of the 4th IEEE Symposium on Bioinformatics and Bioengineering
On the tandem duplication-random loss model of genome rearrangement
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
New Bounds and Tractable Instances for the Transposition Distance
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Perfect Sorting by Reversals Is Not Always Difficult
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Kernel method and linear recurrence system
Journal of Computational and Applied Mathematics
A variant of the tandem duplication — random loss model of genome rearrangement
Theoretical Computer Science
Analytic Combinatorics
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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 2d-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"dd^n, as n-~.