A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 8, 9, 8, 6, 21, 14, 20, 26, 31, 24, 53, 35, 60, 68, 78, 76, 140, 115, 163, 183, 232, 218, 343, 301, 433, 432, 565, 542, 774, 728, 958, 977, 1251, 1220, 1612, 1561, 2053, 2090, 2618, 2609, 3326, 3378

Offset: 0

Author

Gus Wiseman, Feb 16 2022

Keywords

• nonn

Comments

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 distinct, so y is counted under A239455(9).

A partition is Wilf iff it has distinct multiplicities of parts. For example, (2,2,2,1,1,1) has multiplicities (3,3), so is not counted under A098859(9).

The Heinz numbers of these partitions are given by A351294 \ A130091.

Is a(17) = 0 the last zero of the sequence?

Examples

			The a(9) = 1 through a(18) = 5 partitions are (empty columns not shown):
  n=9:      n=12:       n=15:         n=16:       n=18:
  --------------------------------------------------------------
  (222111)  (333111)    (333222)      (33331111)  (444222)
            (22221111)  (444111)                  (555111)
                        (2222211111)              (3322221111)
                                                  (32222211111)
                                                  (222222111111)
		

Crossrefs

Wilf partitions are counted by A098859, ranked by A130091.

Look-and-Say partitions are counted by A239455, ranked by A351294.

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

Non-Look-and-Say partitions are counted by A351293, ranked by A351295.

A000569 = number of graphical partitions, complement A339617.

A032020 = number of binary expansions with all distinct run-lengths.

A044813 = numbers whose binary expansion has all distinct run-lengths.

A225485/A325280 = frequency depth, ranked by A182850/A323014.

A329738 = compositions with all equal run-lengths.

A329739 = compositions with all distinct run-lengths

A351013 = compositions with all distinct runs.

A351017 = binary words with all distinct run-lengths, for all runs A351016.

A351292 = patterns with all distinct run-lengths, for all runs A351200.

Cf. A000041, A008284, A018783, A047966, A181819, A182857, A238130, A305563, A319149, A351203, A351204, A351290.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&Select[Permutations[#], UnsameQ@@Length/@Split[#]&]!={}&]],{n,0,15}]

Formula

a(n) = A239455(n) - A098859(n). Here we assume A239455(0) = 1.

Extensions

More terms from Jinyuan Wang, Feb 14 2025