A105489 Number of partitions of {1...n} containing 3 detached pairs of consecutive integers, i.e., partitions in which only 1- or 2-strings of consecutive integers can appear in a block and there are exactly three 2-strings.

2, 20, 150, 1040, 7105, 49112, 347760, 2537640, 19135875, 149285400, 1205088742, 10062575068, 86859191510, 774456785200, 7126496659960, 67617733760064, 660932425168071, 6649326113764980, 68793130453044760, 731299516881396540

Offset: 6

Author

Augustine O. Munagi, Apr 10 2005

Keywords

• easy
• nonn

Comments

Number of partitions enumerated by A105480 in which the maximal length of consecutive integers in a block is 2.

With offset 3t, number of partitions of {1...N} containing 3 detached strings of t consecutive integers, where N = n + 3j, t = 2 + j, j = 0, 1, 2, ..., i.e., partitions of {1,..,N} in which only v-strings of consecutive integers can appear in a block, where v=1 or v=t and there are exactly three t-strings.

Examples

			a(6) = 2 because the partitions of {1,2,3,4,5,6} with 3 detached pairs of consecutive integers are 12/34/56, 1256/34.
		

Links

• A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.

Crossrefs

Cf. A105480, A105485, A105488, A105490.

Programs

Maple

seq(binomial(n-3,3)*combinat[bell](n-4),n=6..25);
a:=n->sum(numbcomb (n,2)*bell(n)/3, j=0..n): seq(a(n), n=2..21); # Zerinvary Lajos, Apr 25 2007

Formula

a(n) = binomial(n-3, 3)*Bell(n-4), which is the case r=3 in the general case of r pairs, d(n,r) = binomial(n-r, r)*Bell(n-r-1), which is the case t=2 of the general formula d(n,r,t) = binomial(n-r*(t-1), r)*Bell(n-r*(t-1)-1).