A105490 Number of partitions of {1...n} containing 4 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 four 2-strings.

5, 75, 780, 7105, 61390, 521640, 4440870, 38271750, 335892150, 3012721855, 27672081437, 260577574530, 2516984551900, 24942738309860, 253566501600240, 2643729700672284, 28259635983501165, 309569087038701420

Offset: 8

Author

Augustine O. Munagi, Apr 10 2005

Keywords

• easy
• nonn

Comments

Number of partitions enumerated by A105481 in which the maximal length of consecutive integers in a block is 2.

With offset 4t, number of partitions of {1...N} containing 4 detached strings of t consecutive integers, where N = n + 4j, 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 four t-strings.

Examples

			a(8) = 5 because the partitions of {1,...,8} with 4 detached pairs of consecutive integers are 1256/3478, 1256/34/78, 12/3478/56, 1278/34/56, 12/34/56/78.
		

Links

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

Crossrefs

Cf. A105481, A105489, A105491, A105486.

Programs

Maple

seq(binomial(n-4, 4)*combinat[bell](n-5), n=8..28);
with(combinat): a:=n->sum(numbcomb(n-5, 3)*bell(n-5)/4, j=0..n-5): seq(a(n), n=8..28); # Zerinvary Lajos, Apr 25 2007

Formula

a(n) = binomial(n-4, 4)*Bell(n-5), which is the case r=4 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).