A192100 Table read by rows of numbers of unordered pairs of partitions of n-element set that have Rand distance k (n>=2, 1 <= k <= n(n-1)/2).

1, 3, 6, 1, 12, 30, 32, 24, 6, 1, 50, 150, 280, 300, 240, 220, 60, 15, 10, 1, 225, 780, 1720, 3360, 3426, 4100, 2400, 2700, 1075, 471, 150, 35, 45, 15, 1, 1092, 4200, 10885, 25200, 42672, 56889, 60165, 57750, 46585, 31374, 24528, 14140, 4725, 1890, 1302, 252, 210, 140, 105, 21, 1

Offset: 1

Author

Frank Ruskey and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jul 28 2011

Keywords

• tabf
• nonn

Comments

The Rand distance of two set partitions is the number of unordered pairs {x,y} such that there is a block in one partition containing both x and y, but x and y are in different blocks in the other partition. Let R(n,k) denote the number of unordered pairs of partitions of a n-element set that have Rand distance k.

The (n,k) entry contains R(n,k) where n is a row number and k is a column number. Rows are of length C(n,2) = n(n-1)/2 and n is in the range [2..7]. Columns are counted from 1.

Examples

			The table starts:
1
3 6 1
12 30 32 24 6 1
50 150 280 300 240 220 60 15 10 1
225 780 1720 3360 3426 4100 2400 2700 1075 471 150 35 45 15 1
...
One of the 300 pairs of partitions of 5-element set having Rand distance 4:
  {1, 2, 3}{4, 5}
  {1, 2}{3, 4}{5}
		

References

• Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, Combinatorial algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
• Frank Ruskey, Jennifer Woodcock and Yuji Yamauchi, Counting and computing the Rand and block distances of pairs of set partitions, Journal of Discrete Algorithms, Volume 16, October 2012, Pages 236-248. - From N. J. A. Sloane, Oct 03 2012

Links

• Frank Ruskey, Rows n = 2..13, flattened
• F. Ruskey and J. Woodcock, The Rand and block distances of pairs of set partitions, Combinatorial algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.

Crossrefs

Cf. A105479 (first column), A193317.