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A192100
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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).
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6
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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
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LINKS
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EXAMPLE
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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}
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CROSSREFS
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KEYWORD
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tabf,nonn
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AUTHOR
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Frank Ruskey and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jul 28 2011
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STATUS
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approved
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