Geodesic
Cyclic permutation of elements in one-dimensional array
Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 145-149.

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In this paper, we obtain an expression for the smallest number $J(N,K)$ of pairwise permutations of elements in an one-dimensional $N$-element array for resulting in the array being cyclically shifted $k$ positions. An algorithm implementing this $k$-cyclic permutation is also constructed. For an arbitrary integer $i$, $0\le i$, let $\omega(i)$ denote the smallest integer for which $f^{(\omega(i))}(i)\ge i$, where $f(i)=i-k$ if $i\ge k$, and $f(i)=N(1+[k/N])-k+i$ otherwise. Then the smallest number $J(N,K)$ equals the cardinality of the set $\{i\colon N>i\ge0\ \\ g(i)> i\}$, where the map $g\colon\{0,\dots,N-1\}\to\{0,\dots,N-1\}$ is given by the rule $g(i)=f^{(\omega(i))}(i)$ $(0\le i$.
Keywords: one-dimensional array, pairwise permutation of array elements.
Mots-clés : $k$-cyclic permutation 
@article{PDMA_2017_10_a56,
     author = {V. V. Gotsulenko},
     title = {Cyclic permutation of elements in one-dimensional array},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {145--149},
     publisher = {mathdoc},
     number = {10},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2017_10_a56/}
}
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V. V. Gotsulenko. Cyclic permutation of elements in one-dimensional array. Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 145-149. http://geodesic.mathdoc.fr/item/PDMA_2017_10_a56/

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