A006841 Permutation arrays of period n.

1, 1, 1, 2, 4, 10, 28, 127, 686, 4975, 42529, 420948, 4622509, 55670332, 726738971, 10217376792, 153848448652, 2470073249960, 42120966152815, 760282326662191, 14481561464994821, 290289454462745374, 6108699653117045614

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Also superpositions of cycles of order n of the groups S_2 and D_n. - Sean A. Irvine, Oct 25 2017

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 171.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• A. P. Street and R. Day, Sequential binary arrays II: Further results on the square grid, pp. 392-418 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.

Links

• M. Engelhardt, Java program
• V. Meally, Letter to N. J. A. Sloane, Feb 1975
• E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143. (See Table 2, but note errors.)
• A. P. Street and R. Day, Sequential binary arrays II: Further results on the square grid, pp. 392-418 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982. (Annotated scanned copy)
• Venta Terauds, J. Sumner, Circular genome rearrangement models: applying representation theory to evolutionary distance calculations, arXiv preprint arXiv:1712.00858 [q-bio.PE], 2017.

Crossrefs

Cf. A061417.

Formula

Asymptotic behavior: The n-th term T(n) is always larger than n! / (8*n^2) = (n-1)! / 8n; for large n, it is approximated by that value. Stated as formula: T(n) > (n-1)! / 8n; lim 8n * T(n) / (n-1) = 1 as n tends to infinity.

Extensions

Terms for n=1..8 from A. P. Street and R. Day; other terms computed by Matthias Engelhardt. For n=9..12, he used a program which shifts, rotates and mirrors permutations. Terms for n=13..29 computed with a Java program implementing the formulas.