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Number of complete mappings f(x) of the cyclic group Z_{2n+1} such that -f(-x)=f.
2

%I #9 Aug 04 2024 20:32:58

%S 1,1,3,5,21,69,319,1957,12513,85445,656771,5591277,51531405,509874417,

%T 5438826975,62000480093,752464463029,9685138399785,131777883431119

%N Number of complete mappings f(x) of the cyclic group Z_{2n+1} such that -f(-x)=f.

%C A complete mapping of a cyclic group (Zn,+) is a permutation f(x) of Zn such that f(0)=0 and that f(x)-x is also a permutation.

%C a(n) is the number of complete mappings fixed under rotation R180 where R180(f)(x)=-f(-x). This sequence (n) equals TSQ_R180(n), the number of solutions of the toroidal n-queen problem fixed under rotation R180. A solution of toroidal-semi n-queen problem is a permutation f(x) of Zn such that f(x)-x is also a permutation.

%D Y. P. Shieh, "Partition strategies for #P-complete problems with applications to enumerative combinatorics", PhD thesis, National Taiwan University, 2001.

%D Y. P. Shieh, J. Hsiang and D. F. Hsu, "On the enumeration of Abelian k-complete mappings", vol. 144 of Congressus Numerantium, 2000, pp. 67-88.

%H Y. P. Shieh, <a href="http://turing.csie.ntu.edu.tw/~arping/cm">Cyclic complete mappings counting problems</a>

%e f(x)=6x in (Z7,+) is a complete mapping of Z7 since f(0)=0 and f(x)-x (=5x) is also a permutation of Z7. R180(f)(x)=-f(-x) (=6x). So f(x) is fixed under R180.

%K nonn

%O 0,3

%A J. Hsiang, D. F. Hsu and Y. P. Shieh (arping(AT)turing.csie.ntu.edu.tw), Jun 03 2002

%E Offset corrected by _Sean A. Irvine_, Aug 04 2024