A001192 Number of full sets of size n.

1, 1, 1, 2, 9, 88, 1802, 75598, 6421599, 1097780312, 376516036188, 258683018091900, 355735062429124915, 978786413996934006272, 5387230452634185460127166, 59308424712939278997978128490, 1305926814154452720947815884466579

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

A set x is full if every element of x is also a subset of x.

Equals the subpartitions of Eulerian numbers (A000295(n)=2^n-n-1); see A115728 for the definition of subpartitions of a partition. - Paul D. Hanna, Jul 03 2006

Also number of transitive rooted identity trees with n branches. - Gus Wiseman, Dec 21 2016

Examples

			Examples of full sets are 0 := {}, 1 := {0}, 2 := {1,0}, 3a := {2,1,0}, 3b := { {1}, 1, 0}, 4a := { 3a, 2, 1, 0 }.
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 123, Problem 20.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..50
• Bojan Bašić, Paul Ellis, Dana C. Ernst, Danijela Popović, and Nándor Sieben, Categories of impartial rulegraphs and gamegraphs, arXiv:2312.00650 [math.CO], 2023. See p. 20.
• Alberto Casagrande, Carla Piazza, and Alberto Policriti, Is hyper-extensionality preservable under deletions of graph elements?, Elec. Notes Theor. Comp. Sci. (2016) Vol. 322, 103-118.
• Richard Peddicord, The number of full sets with n elements, Proc. Amer. Math. Soc., 13 (1962), 825-828.
• John Riordan, Letter to N. J. A. Sloane, Jul. 1970
• Alexandru Ioan Tomescu, Sets as Graphs, Ph. D. Thesis, Università degli Studi di Udine, Dipartimento di Matematica e Informatica, Dottorato di Ricerca in Informatica, Dec. 2011.
• Stephan Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).
• Gus Wiseman, Transitive rooted identity trees with n=5 branches

Crossrefs

Cf. A000295, A004111, A115728, A182161, A279861, A279863.

Programs

Maple

A001192 := proc(n) option remember: if(n=0)then return 1: fi: return add((-1)^(n-k-1)*binomial(2^k-k,n-k)*procname(k), k=0..n-1); end: seq(A001192(n), n=0..16); # Nathaniel Johnston, Apr 18 2012

Mathematica

max = 16; f[x_] := Sum[a[n]*(x^n/(1+x)^2^n), {n, 0, max}] - 1; cc = CoefficientList[ Series[f[x], {x, 0, max}], x]; Table[a[n], {n, 0, max}] /. First[ Solve[ Thread[cc == 0]]] (* Jean-François Alcover, Nov 02 2011, after Vladeta Jovovic *)

PARI

{a(n)=polcoeff(x^n-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(2^k-k-1) ), n)} \\ Paul D. Hanna, Jul 03 2006

Formula

1 = Sum_{n>=0} a(n)*x^n/(1+x)^(2^n). E.g., 1 = 1/(1+x) + 1*x/(1+x)^2 + 1*x^2/(1+x)^4 + 2*x^3/(1+x)^8 + 9*x^4/(1+x)^16 + 88*x^5/(1+x)^32 + 1802*x^6/(1+x)^64 + ... . - Vladeta Jovovic, May 26 2005

Equivalently, a(n) = (-1)^n*C(2^n+n-1, n) - Sum_{k=0..n-1} a(k)*(-1)^(n-k)*C(2^n+2^k+n-k-1, n-k). - Paul D. Hanna, May 26 2005

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^(2^n-n-1) = 1*(1-x)^0 + 1*x*(1-x)^0 + 1*x^2*(1-x)^1 + 2*x^3*(1-x)^4 + 9*x^3*(1-x)^11 + ... + a(n)*x^n*(1-x)^(2^n-n-1) + ... . - Paul D. Hanna, Jul 03 2006

Extensions

More terms from Ryan Propper, Jun 13 2005