cp's OEIS Frontend

A275549 Number of classes of endofunctions of [n] under reversal.

%I A275549 #14 Sep 13 2019 20:24:12
%S A275549 1,1,3,18,136,1625,23436,412972,8390656,193739769,5000050000,
%T A275549 142656721086,4458051717120,151437584670385,5556003465485760,
%U A275549 218946946471875000,9223372039002259456,413620131002462320337,19673204037747448432896,989209827833222327690890
%N A275549 Number of classes of endofunctions of [n] under reversal.
%C A275549 f and g are in the same class if function g(i) = f(n+1-i) for all i.
%C A275549 Decomposition by class size
%C A275549 .
%C A275549 n    1      2
%C A275549 ---------------
%C A275549 1    1      0
%C A275549 2    2      1
%C A275549 3    9      9
%C A275549 4    16     120
%C A275549 5    125    1500
%C A275549 6    216    23220
%C A275549 7    2401   410571
%C A275549 .
%C A275549 Demonstration for the formula: the classes are either of size 1 or 2.
%C A275549 The classes of size 1 is for functions invariant by reversal. They are specified by half their values, including one more if n is odd. Their number is n^(ceiling(n/2)).
%C A275549 So the number of classes under this symmetry is half (the number of functions + the number of classes of size 1).
%C A275549 a(n) is the number of unoriented length n strings with a maximum of n colors. - _Andrew Howroyd_, Sep 13 2019
%H A275549 Andrew Howroyd, <a href="/A275549/b275549.txt">Table of n, a(n) for n = 0..200</a>
%F A275549 a(n) = (n^n+n^ceiling(n/2))/2.
%o A275549 (PARI) a(n) = {(n^n + n^((n+1)\2))/2} \\ _Andrew Howroyd_, Sep 13 2019
%Y A275549 Main diagonal of A277504.
%Y A275549 Cf. A000312 All endofunctions
%Y A275549 Cf. A000169 Classes under translation mod n
%Y A275549 Cf. A001700 Classes under sort
%Y A275549 Cf. A056665 Classes under rotation
%Y A275549 Cf. A168658 Classes under complement to n+1
%Y A275549 Cf. A130293 Classes under translation and rotation
%Y A275549 Cf. A081721 Classes under rotation and reversal
%Y A275549 Cf. A275550 Classes under reversal and complement
%Y A275549 Cf. A275551 Classes under translation and reversal
%Y A275549 Cf. A275552 Classes under translation and complement
%Y A275549 Cf. A275553 Classes under translation, complement and reversal
%Y A275549 Cf. A275554 Classes under translation, rotation and complement
%Y A275549 Cf. A275555 Classes under translation, rotation and reversal
%Y A275549 Cf. A275556 Classes under translation, rotation, complement and reversal
%Y A275549 Cf. A275557 Classes under rotation and complement
%Y A275549 Cf. A275558 Classes under rotation, complement and reversal
%Y A275549 Cf. A078707 Endofunctions symmetric around their middle (stable by reversal).
%K A275549 nonn,easy
%O A275549 0,3
%A A275549 _Olivier Gérard_, Aug 01 2016