A127912 Number of nonisomorphic disconnected mappings (or mapping patterns) from n points to themselves; number of disconnected endofunctions.

0, 1, 3, 10, 27, 79, 218, 622, 1753, 5007, 14274, 40954, 117548, 338485, 975721, 2817871, 8146510, 23581381, 68322672, 198138512, 575058726, 1670250623, 4854444560, 14117859226, 41081418963, 119606139728

Offset: 0

Author

Jonathan Vos Post, Feb 06 2007

Keywords

• easy
• nonn

Comments

Number of endofunctions on n points whose functional digraphs (with loops allowed) are nontrivially the directed sum of two or more digraphs of endofunctions.

Examples

			a(0) = 0, as the null digraph is formally neither connected nor disconnected.
a(1) = 0, as the unique endofunction on one point is the identity function on one value and is connected.
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.
a(3) = A001372(3) - A002861(3) = 7 - 4 = 3.
a(4) = A001372(4) - A002861(4) = 19 - 9 = 10.
a(5) = A001372(5) - A002861(5) = 47 - 20 = 27.
a(6) = 130 - 51 = 79.
a(7) = 343 - 125 = 218.
a(8) = 951 - 329 = 622.
a(9) = 2615 - 862 = 1753.
a(10) = 7318 - 2311 = 5007.
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.6.
• R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.399 and 41.401.

Links

• N. G. de Bruijn and D. A. Klarner, Multisets of aperiodic cycles, SIAM J. Algebraic Discrete Methods 3 (1982), no. 3, 359--368. MR0666861(84i:05008).
• Eric Weisstein's World of Mathematics, Functional Graph.

Crossrefs

Cf. A000081, A000273, A001372, A002861, A003027, A003085, A062738, A116950, A126285, A127909-A127915.

Formula

a(n) = A001372(n) - A002861(n).