A124104 Sum of the Rand distance between all pairs of set partitions of {1, 2, ... n}.

0, 2, 36, 600, 11100, 235560, 5746524, 160252456, 5069446560, 180479494440, 7177165063750, 316636751823480, 15401586272510880, 821382267765103590, 47788292465454829260, 3019446671476746981600, 206339807951889894605488

Offset

1,2

Comments

From Joshua Zucker, Dec 21 2006: (Start)

Given a partition of {1, ..., n}, look at a pair of elements. If the two elements are in the same block of the partition, they're called "co-clustered". The Rand distance between two partitions then counts the pairs that are co-clustered in exactly one of the two partitions. The Rand index is found by dividing the Rand distance by (n choose 2).

Example: The distance from 12 3 4 to 1 234 is 4 because of the four pairs 12 (in the first partition but not the second) and 23, 24, 34 (in the second partition but not the first). The maximal distance of 6 is attained by 1 2 3 4 and 1234. The Rand distance has some nice properties, satisfies the triangle inequality and there are linear-time algorithms for computing it. (End)

Links

Table of n, a(n) for n=1..17.

V. Filkov and S. Skiena, Integrating microarray data by consensus clustering, (see also doi: 10.1142/S0218213004001867 or 10.1109/TAI.2003.1250220).

A. Goder and V. Filkov, Consensus clustering algorithms: Comparison and refinement, Proceedings of the Tenth Workshop on Algorithm Engineering and Experiments, 2008, 109-117.

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

Formula

a(n) = 2 * binomial(n,2) * Bell(n-1) * (Bell(n) - Bell(n-1)).

a(n) ~ n*LambertW(n)*Bell(n)^2 * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

Example

E.g. a(2) = 2 = 1 + 1 + 0 + 0 because the distance from 1,2 to 12 is 1 (and vice versa) and the distance from 1,2 to 1,2 or 12 to 12 is 0.

Mathematica

Table[2 Binomial[n, 2]*BellB[n - 1] (BellB[n] - BellB[n - 1]), {n, 17}] (* Michael De Vlieger, Apr 16 2015 *)

Crossrefs

Equals twice A193317.

Sequence in context: A141132 A064030 A228790 * A262973 A207832 A112036

Adjacent sequences: A124101 A124102 A124103 * A124105 A124106 A124107

Keyword

nonn

Author

Andrey Goder, Dec 12 2006, Feb 20 2007

Status

approved