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A239454 Array: row n shows the Look-and-Say partitions of n >=1, in Mathematica order. 2
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, 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, 1, 3, 2, 2, 3, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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).
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].
LINKS
EXAMPLE
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.
The 11 partitions of 6 generate Look-and-Say partitions as follows:
6 -> 111111
51 -> 111111
42 -> 111111
411 -> 21111
33 -> 222
321 -> 111111
3111 -> 3111
222 -> 33
2211 -> 222
21111 -> 411
111111 -> 6,
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:
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
6 4 1 1 3 3 3 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1
MATHEMATICA
LS[part_List] := Reverse[Sort[Flatten[Map[Table[#[[2]], {#[[1]]}] &, Tally[part]]]]]; LS[n_Integer] := #[[Reverse[Ordering[PadRight[#]]]]] &[DeleteDuplicates[Map[LS, IntegerPartitions[n]]]];
TableForm[t = Map[LS[#] &, Range[10]]](*A239454, array*)
Flatten[t](*A239454, sequence*)
Map[Length[LS[#]] &, Range[30]](*A239455*)
(* Peter J. C. Moses, Mar 18 2014 *)
CROSSREFS
Sequence in context: A182980 A244051 A207974 * A108888 A124021 A109626
KEYWORD
nonn,tabf,easy
AUTHOR
STATUS
approved

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Last modified July 17 22:17 EDT 2024. Contains 374377 sequences. (Running on oeis4.)