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Number of partitions of {1...n} containing 2 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 two 2-strings.
11

%I #29 May 11 2019 10:10:52

%S 1,6,30,150,780,4263,24556,149040,951615,6378625,44785620,328660566,

%T 2515643767,20044428810,165955025400,1425299331992,12678325080012,

%U 116635133853189,1108221018960830,10862073229428120,109694927532209481,1140199081827172719

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

%C Number of partitions enumerated by A105479 in which the maximal length of consecutive integers in a block is 2.

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

%C Equals the minimum of the sum of the Rand distances over all A000110(n) set partitions of n elements. E.g. a(3) = 6 because over the 5 set partitions of {1, 2, 3} the sum of Rand distances from {{1}, {2}, {3}} to the rest is 6. - Andrey Goder (andy.goder(AT)gmail.com), Dec 08 2006

%C a(n+3) = A000110(n) * A000217(n) = Sum_{k=1..n} A285362(n,k) is the sum of the entries in all set partitions of [n]. - _Alois P. Heinz_, Apr 16 2017

%H Alois P. Heinz, <a href="/A105488/b105488.txt">Table of n, a(n) for n = 4..577</a>

%H A. O. Munagi, <a href="http://dx.doi.org/10.1155/IJMMS.2005.451">Set Partitions with Successions and Separations</a>, IJMMS 2005:3 (2005), 451-463.

%H W. Rand, <a href="http://www.jstor.org/stable/2284239">Objective criteria for the evaluation of clustering methods</a>, J. Amer. Stat. Assoc., 66 (336): 846-850, 1971.

%F a(n) = binomial(n-2, 2)*Bell(n-3), which is the case r = 2 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(5)=6 because the partitions of {1,2,3,4,5} with 2 detached pairs of consecutive integers are 145/23,125/34,1245/3,12/34/5,12/3/45,1/23/45.

%p seq(binomial(n-2,2)*combinat[bell](n-3),n=4..28);

%t a[n_] := Binomial[n-2, 2]*BellB[n-3];

%t Table[a[n], {n, 4, 25}] (* _Jean-François Alcover_, May 11 2019 *)

%Y Cf. A000110, A000217, A052889, A105479, A105489, A105484, A271023, A285362.

%K easy,nonn

%O 4,2

%A _Augustine O. Munagi_, Apr 10 2005