Sorting Permutations with Fixed Pinnacle Set
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We give a positive answer to a question raised by Davis et al. ( Discrete Mathematics 341, 2018), concerning permutations with the same pinnacle set. Given π∈ S_n, a pinnacle of π is an element π_i (i≠ 1,n) such that π_i-1<π_i>π_i+1. The question is: given π,π'∈ S_n with the same pinnacle set S, is there a sequence of operations that transforms π into π' such that all the intermediate permutations have pinnacle set S? We introduce balanced reversals, defined as reversals that do not modify the pinnacle set of the permutation to which they are applied. Then we show that π may be sorted by balanced reversals (i.e. transformed into a standard permutation _S), implying that π may be transformed into π' using at most 4n-2min{p,3} balanced reversals, where p=|S|≥ 1. In case p=0, at most 2n-1 balanced reversals are needed.
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