|
%I #8 Nov 06 2017 08:57:52
%S 0,1,3,10,27,79,218,622,1753,5007,14274,40954,117548,338485,975721,
%T 2817871,8146510,23581381,68322672,198138512,575058726,1670250623,
%U 4854444560,14117859226,41081418963,119606139728
%N Number of nonisomorphic disconnected mappings (or mapping patterns) from n points to themselves; number of disconnected endofunctions.
%C Number of endofunctions on n points whose functional digraphs (with loops allowed) are nontrivially the directed sum of two or more digraphs of endofunctions.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.6.
%D R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.399 and 41.401.
%D N. G. de Bruijn and D. A. Klarner, Multisets of aperiodic cycles, SIAM J, Algeb. Discrete Meth., 3 (1982), 359-368.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FunctionalGraph.html">Functional Graph.</a>
%F a(n) = A001372(n) - A002861(n).
%e a(0) = 0, as the null digraph is formally neither connected nor disconnected.
%e a(1) = 0, as the unique endofunction on one point is the identity function on one value and is connected.
%e a(2) = 1, as there are 3 endofunctions on two points, two of which are "prime endofunctions" and one of which is the direct sum of two copies of the unique endofunction on one point, namely two points-with-loops, or the identity function on two values; 3 - 2 = 1.
%e a(3) = A001372(3) - A002861(3) = 7 - 4 = 3.
%e a(4) = A001372(4) - A002861(4) = 19 - 9 = 10.
%e a(5) = A001372(5) - A002861(5) = 47 - 20 = 27.
%e a(6) = 130 - 51 = 79.
%e a(7) = 343 - 125 = 218.
%e a(8) = 951 - 329 = 622.
%e a(9) = 2615 - 862 = 1753.
%e a(10) = 7318 - 2311 = 5007.
%Y Cf. A000081, A000273, A001372, A002861, A003027, A003085, A062738, A116950, A126285, A127909-A127915.
%K easy,nonn
%O 0,3
%A _Jonathan Vos Post_, Feb 06 2007
| 