Short Proofs for Cut-and-Paste Sorting of Permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of $[n]$ using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of $[n]$ can be transformed to the identity in at most \flr{2n/3} such moves and that some permutations require at least \flr{n/2} moves.Comment: 7 pages, 2 figure

Bounds for Permutation Arrays under Kendall Tau Metric

Permutation arrays under the Kendall-$\tau$ metric have been considered for error-correcting codes. Given $n$ and $d\in [1..\binom{n}{2}]$, the task is to find a large permutation array of permutations on $n$ symbols with pairwise Kendall-$\tau$ distance at least $d$. Let $P(n,d)$ denote the maximum size of any permutation array of permutations on $n$ symbols with pairwise Kendall-$\tau$ distance $d$. New algorithms and several theorems are presented, giving improved lower bounds for $P(n,d)$. Also, $(n,m,d)$-arrays are defined, which are permutation arrays on n symbols with Kendall-$\tau$ distance d, with the restriction that symbols {1...(n-m)} appear in increasing order. Let $P(n,m,d)$ denote the maximum size of any $(n,m,d)$-array. For example, (n,m,d)-arrays are useful for recursively computing lower bounds for $P(n,d)$. Lower and upper bounds are given for $P(n.m,d)$

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The pancake problem, which has attracted considerable attention [10, 12, 9, 15], concerns the number of prefix reversals or flips needed to sort the elements of an arbitrary permutation. The number of prefix reversals to sort permutations is also the diameter of the often studied n-dimensional Pancake network P n [3, 4, 5, 8, 14, 16]. We consider restricted pancake problems, for example when only 3 of the possible n- 1 flips are allowed. Let f i denote a flip of size i. Each flip is itself a permutation. For example, a flip of size 4, i. e., f 4, on eight symbols has the effect of changing, say, 3 5 1 2 4 6 8 7 into 2 1 5 3 4 6 8 7. f 4 is the permutation 4 3 2 1 5 6 7 8. We investigate sets of permutations corresponding to flips as generators of the symmetric group S n. Let n be the number of symbols in a permutation. We consider sets with either a constant number of generators (i. e., flips) or with log 2 n generators. In special interesting cases, the corresponding Cayley networks, defined by a given set of generators and a given group of permutations, are explored. Specifically, we investigate two special families of networks: 1) The Subcube network, for n = 2 n k, defined by the log2 n generators in the set {f2, f4, f8 … fn}