Abstract
We suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants. We then generalize the class of skew Young tableaux under consideration; this allows in particular to discover some presumably new results concerning Eulerian numbers.
Highlights
This article deals with minimal permutations with d descents
Genomes are represented by permutations, and minimal permutations with d = 2p descents are the basis of excluded patterns that describes the class of permutations that can be obtained from the identity with cost at most p
In [MY], further results on the enumeration of minimal permutations with d descents have been obtained using multivariate generating functions, allowing in particular to derive a closed formula enumerating those of size 2d − 1 as well as some asymptotic results
Summary
This article deals with minimal permutations with d descents ( called d-minimal permutations here) This family of permutations has been introduced in [BoRo] in the study of the whole genome duplication-random loss model of genome rearrangement. In this work we offer an alternative approach for the enumeration of minimal permutations with d descents, making extensive use of a bijection between these permutations and a family of skew Young tableaux. This gives a general formula for the number pd+k,d of minimal permutations with d descents and of size d + k, as a sum of determinants of matrices (Theorem 4.2).
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