%I #20 Feb 22 2018 16:34:40
%S 1,2,1,1,3,1,1,1,4,2,2,2,1,1,1,1,1,1,5,3,1,1,2,2,1,2,1,1,1,1,1,1,1,1,
%T 6,4,1,1,3,3,3,1,1,1,2,2,2,2,1,1,1,1,1,1,1,1,1,1,7,5,1,1,4,1,1,1,3,3,
%U 1,3,2,2,3,1,1,1,1,2,2,2,1,2,2,1,1
%N Array: row n shows the Look-and-Say partitions of n >=1, in Mathematica order.
%C Suppose that p = x(1) >= x(2) >= … >= x(k) is a partition of n. Let y(1) > y(2) > … > y(h) be the distinct parts of p, and let m(i) be the multiplicity of y(i) for 1 <= i <= h. Then we can "look" at p as "m(1) y(1)'s and m(2) y(2)'s and … m(h) y(h)'s". Reversing the m's and y's, we can then "say" the Look-and-Say partition of p, denoted by LS(p). The name "Look-and-Say" follows the example of Look-and-Say integer sequences (e.g., A005150). As p ranges through the partitions of n, LS(p) ranges through all the Look-and-Say partitions of n. The number of these is A239455(n).
%C The Look-and-Say array is distinct from the Wilf array, described at A098859; for example, the number of Look-and-Say partitions of 9 is A239455(9) = 16, whereas the number of Wilf partitions of 9 is A098859(9) = 15. The Look-and-Say partition of 9 which is not a Wilf partitions of 9 is [2,2,2,1,1,1].
%H Clark Kimberling, <a href="/A239454/b239454.txt">Table of n, a(n) for n = 1..10000</a>
%e Look at the partition p = 5442111 as 1*5 + 2*4 + 1*2 + 3*1, which equals 5*1 + 4*2 + 3*2 + 1*3, for which we can say LS(p) = 3221111111.
%e The 11 partitions of 6 generate Look-and-Say partitions as follows:
%e 6 -> 111111
%e 51 -> 111111
%e 42 -> 111111
%e 411 -> 21111
%e 33 -> 222
%e 321 -> 111111
%e 3111 -> 3111
%e 222 -> 33
%e 2211 -> 222
%e 21111 -> 411
%e 111111 -> 6,
%e so that row 6 results from arranging in Mathematica order the partitions 111111, 21111, 222, 3111, 33, 411, 6, as 6, 411, 33, 3111, 222, 21111, 111111. Following are the first 6 rows:
%e 1
%e 2 1 1
%e 3 1 1 1
%e 4 2 2 2 1 1 1 1 1 1
%e 5 3 1 1 2 2 1 2 1 1 1 1 1 1 1 1
%e 6 4 1 1 3 3 3 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1
%t LS[part_List] := Reverse[Sort[Flatten[Map[Table[#[[2]], {#[[1]]}] &, Tally[part]]]]]; LS[n_Integer] := #[[Reverse[Ordering[PadRight[#]]]]] &[DeleteDuplicates[Map[LS, IntegerPartitions[n]]]];
%t TableForm[t = Map[LS[#] &, Range[10]]](*A239454,array*)
%t Flatten[t](*A239454,sequence*)
%t Map[Length[LS[#]] &, Range[30]](*A239455*)
%t (* Peter J. C. Moses, Mar 18 2014 *)
%Y Cf. A239455, A005150, A098859.
%K nonn,tabf,easy
%O 1,2
%A _Clark Kimberling_ and _Peter J. C. Moses_, Mar 19 2014