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A105491 Number of partitions of {1...n} containing 5 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 five 2-strings. 3

%I #5 Oct 03 2013 09:34:33

%S 15,312,4263,49112,521640,5329044,53580450,537427440,5422899339,

%T 55344162874,573270663966,6040762924560,64851119605636,

%U 709986204480672,7931189102016852,90430835147203728,1052534895931584828

%N Number of partitions of {1...n} containing 5 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 five 2-strings.

%C Number of partitions enumerated by A105482 in which the maximal length of consecutive integers in a block is 2.

%C With offset 5t, number of partitions of {1,...,N} containing 5 detached strings of t consecutive integers, where N=n+5j, 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 five t-strings.

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

%H A. O. Munagi, <a href="http://www.emis.de/journals/HOA/IJMMS/2005/3451.pdf">Set Partitions with Successions and Separations</a>,IJMMS 2005:3 (2005),451-463.

%F a(n)=binomial(n-5, 5)*Bell(n-6), which is the case r=5 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)*B(n-r*(t-1)-1).

%e a(10)=15; the enumerated 15 partitions of {1,...,10} with 5 detached pairs of consecutive integers include (1,2,5,6,9,10)(3,4,7,8) and (1,2,9,10)(3,4,7,8)(5,6).

%p seq(binomial(n-5,5)*combinat[bell](n-6),n=10..30);

%Y Cf. A105482, A105490, A105487.

%K easy,nonn

%O 10,1

%A _Augustine O. Munagi_, Apr 10 2005

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