A239455 Number of Look-and-Say partitions of n; see Comments.

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 28, 33, 45, 55, 65, 83, 105, 121, 155, 180, 217, 259, 318, 362, 445, 512, 614, 707, 850, 958, 1155, 1309, 1543, 1754, 2079, 2327, 2740, 3085, 3592, 4042, 4699, 5253, 6093, 6815, 7839, 8751, 10069, 11208, 12832, 14266, 16270

Offset

0,3

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 partition of 9 is [2,2,2,1,1,1].

Theorem: a partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are all distinct, so y is counted under a(9). - Gus Wiseman, Feb 17 2022

Links

Table of n, a(n) for n=0..51.

Example

The 11 partitions of 6 generate 7 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 a(6) counts these 7 partitions: 111111, 21111, 222, 3111, 33, 411, 6.

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[25]](*A239455*)
(* Peter J. C. Moses, Mar 18 2014 *)
Table[Length[Select[IntegerPartitions[n], Select[Permutations[#], UnsameQ@@Length/@Split[#]&]!={}&]], {n, 0, 15}] (* Gus Wiseman, Feb 17 2022, a(0)=1 *)

Crossrefs

These include all Wilf partitions, counted by A098859, ranked by A130091.

These partitions are listed by A239454 in graded reverse-lex order.

Non-Wilf partitions are counted by A336866, ranked by A130092.

A variant for runs is A351204, complement A351203.

The complement is counted by A351293, ranked by A351295.

These partitions are ranked by A351294.

The non-Wilf case is counted by A351592.

A000569 = graphical partitions, complement A339617.

A181819 = Heinz number of the prime signature of n (prime shadow).

A329738 = compositions with all equal run-lengths.

Counting words with all distinct run-lengths:

- A032020 = binary expansions, for runs A351018, ranked by A044813.

- A329739 = compositions, for runs A351013, ranked by A351596.

- A351017 = binary words, for runs A351016.

- A351292 = patterns, for runs A351200.

Cf. A000041, A008284, A047966, A182857, A225485, A297770, A304660, A305563, A329746, A351201, A351202, A351291.

Sequence in context: A333190 A098859 A364345 * A362610 A363260 A195012

Adjacent sequences: A239452 A239453 A239454 * A239456 A239457 A239458

Keyword

nonn

Author

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

Status

approved