4,2

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

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.

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

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

W. Rand. Objective criteria for the evaluation of clustering methods. J. Amer. Stat. Assoc., 66 (336): 846-850, 1971.

A. O. Munagi, Set Partitions with Successions and Separations,IJMMS 2005:3 (2005),451-463.

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).

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.

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

with (combinat): [seq (bell(n)*stirling2(n+1, n), n=1..21)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 18 2007

a:=n->sum(numbcomb (n, 1)*bell(n)/2, j=0..n): seq(a(n), n=1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007

a:=n->sum(sum(bell(n)/2, j=1..n), k=0..n): seq(a(n), n=1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

with(combinat): seq(add(add(bell(n)/2, j=1..n), k=0..n), n=1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2007

Cf. A052889, A105479, A105489, A105484.

Sequence in context: A006818 A006819 A003948 this_sequence A054117 A033132 A022023

Adjacent sequences: A105485 A105486 A105487 this_sequence A105489 A105490 A105491

easy,nonn

A. O. Munagi (amunagi(AT)yahoo.com), Apr 10 2005