A105490 - OEIS

%I #14 Jul 26 2017 10:11:20

%S 5,75,780,7105,61390,521640,4440870,38271750,335892150,3012721855,

%T 27672081437,260577574530,2516984551900,24942738309860,

%U 253566501600240,2643729700672284,28259635983501165,309569087038701420

%N 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.

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

%C 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.

%H A. O. Munagi, <a href="http://www.emis.de/journals/HOA/IJMMS/2005/3451.pdf">Set Partitions with Successions and Separations</a>, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.

%F 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).

%e 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.

%p seq(binomial(n-4, 4)*combinat[bell](n-5), n=8..28);

%p 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

%Y Cf. A105481, A105489, A105491, A105486.

%K easy,nonn

%O 8,1

%A _Augustine O. Munagi_, Apr 10 2005