A239454 Array: row n shows the Look-and-Say partitions of n >=1, in Mathematica order.

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

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

Clark Kimberling, Table of n, a(n) for n = 1..10000

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

Cf. A239455, A005150, A098859.

Sequence in context: A182980 A244051 A207974 * A108888 A124021 A109626

Adjacent sequences: A239451 A239452 A239453 * A239455 A239456 A239457

Keyword

nonn,tabf,easy

Author

Clark Kimberling and Peter J. C. Moses, Mar 19 2014

Status

approved