A094198 Number of ways that n boxes with distinct sizes can contain each other under the condition that each box may contain at most three (themselves possibly nested) boxes. Each box is assumed to be large enough to contain any three smaller boxes.

1, 2, 6, 24, 119, 702, 4795, 37183, 322486, 3091630, 32453172, 370104159, 4555518746, 60182704891, 849245520581, 12746759647944, 202753756944382, 3406596290534764, 60282041591986049, 1120554350714688128

Offset: 1

Author

John W. Layman, May 25 2004

Keywords

• nonn

Comments

If each box may contain at most one (possibly nested) box then the Bell numbers (A000110) are obtained, whereas if each box may contain at most two smaller (possibly nested) boxes then A000772 is obtained and if no restriction is placed on the number of (possible nested) boxes that any box may contain then the factorial numbers (A000142) are obtained. Sequence suggested by an earlier submission of Rick L. Shepherd.

Examples

			a(3)=6, as seen from these arrangements: 112233, 321123, 311223, 211233, 223113, 113223, where xyyx indicates that box x contains box y, etc.
		

Links

• Letong Hong and Rupert Li, Length-Four Pattern Avoidance in Inversion Sequences, arXiv:2112.15081 [math.CO], 2021.

Crossrefs

Cf. A000110, A000142, A000772.

Programs

Mathematica

m = 30; G[_] = 1;
Do[G[x_] = 1 + (1/6) Integrate[G[x]^3 + 3 G[x] + 2, x] + O[x]^m, {m}];
CoefficientList[Exp[G[x] - 1] + O[x]^m, x] Range[0, m - 1]! // Rest (* Jean-François Alcover, Nov 13 2019 *)

Formula

E.g.f.: exp(G(x) - 1), where G(x) be the function that satisfies 6G'(x) = G(x)^3 + 3G(x) + 2 and G(0) = 1. In this case G'(x) is the exponential generating function giving the number of ways to perform the given task if at most 3 boxes may fail to lie in another box. [Joel B. Lewis, Apr 28 2009]

a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x+x^2/2!+x^3/3!)*d/dx. Cf. A000110 and A000772. - Peter Bala, Nov 25 2011

Extensions

a(20) corrected by Jean-François Alcover, Nov 13 2019