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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).

%I A192100 #57 Dec 26 2023 09:57:17
%S A192100 1,3,6,1,12,30,32,24,6,1,50,150,280,300,240,220,60,15,10,1,225,780,
%T A192100 1720,3360,3426,4100,2400,2700,1075,471,150,35,45,15,1,1092,4200,
%U A192100 10885,25200,42672,56889,60165,57750,46585,31374,24528,14140,4725,1890,1302,252,210,140,105,21,1
%N 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).
%C A192100 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.
%C A192100 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.
%D A192100 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.
%D A192100 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
%H A192100 Frank Ruskey, <a href="/A192100/b192100.txt">Rows n = 2..13, flattened</a>
%H A192100 F. Ruskey and J. Woodcock, <a href="http://webhome.cs.uvic.ca/~ruskey/Publications/RandDist/RandDist.html">The Rand and block distances of pairs of set partitions</a>, Combinatorial algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
%e A192100 The table starts:
%e A192100 1
%e A192100 3 6 1
%e A192100 12 30 32 24 6 1
%e A192100 50 150 280 300 240 220 60 15 10 1
%e A192100 225 780 1720 3360 3426 4100 2400 2700 1075 471 150 35 45 15 1
%e A192100 ...
%e A192100 One of the 300 pairs of partitions of 5-element set having Rand distance 4:
%e A192100   {1, 2, 3}{4, 5}
%e A192100   {1, 2}{3, 4}{5}
%Y A192100 Cf. A105479 (first column), A193317.
%K A192100 tabf,nonn
%O A192100 1,2
%A A192100 _Frank Ruskey_ and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jul 28 2011