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

%I A351592 #15 Feb 14 2025 05:22:46
%S A351592 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,
%T A351592 35,60,68,78,76,140,115,163,183,232,218,343,301,433,432,565,542,774,
%U A351592 728,958,977,1251,1220,1612,1561,2053,2090,2618,2609,3326,3378
%N A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).
%C A351592 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).
%C A351592 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).
%C A351592 The Heinz numbers of these partitions are given by A351294 \ A130091.
%C A351592 Is a(17) = 0 the last zero of the sequence?
%F A351592 a(n) = A239455(n) - A098859(n). Here we assume A239455(0) = 1.
%e A351592 The a(9) = 1 through a(18) = 5 partitions are (empty columns not shown):
%e A351592   n=9:      n=12:       n=15:         n=16:       n=18:
%e A351592   --------------------------------------------------------------
%e A351592   (222111)  (333111)    (333222)      (33331111)  (444222)
%e A351592             (22221111)  (444111)                  (555111)
%e A351592                         (2222211111)              (3322221111)
%e A351592                                                   (32222211111)
%e A351592                                                   (222222111111)
%t A351592 Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&Select[Permutations[#], UnsameQ@@Length/@Split[#]&]!={}&]],{n,0,15}]
%Y A351592 Wilf partitions are counted by A098859, ranked by A130091.
%Y A351592 Look-and-Say partitions are counted by A239455, ranked by A351294.
%Y A351592 Non-Wilf partitions are counted by A336866, ranked by A130092.
%Y A351592 Non-Look-and-Say partitions are counted by A351293, ranked by A351295.
%Y A351592 A000569 = number of graphical partitions, complement A339617.
%Y A351592 A032020 = number of binary expansions with all distinct run-lengths.
%Y A351592 A044813 = numbers whose binary expansion has all distinct run-lengths.
%Y A351592 A225485/A325280 = frequency depth, ranked by A182850/A323014.
%Y A351592 A329738 = compositions with all equal run-lengths.
%Y A351592 A329739 = compositions with all distinct run-lengths
%Y A351592 A351013 = compositions with all distinct runs.
%Y A351592 A351017 = binary words with all distinct run-lengths, for all runs A351016.
%Y A351592 A351292 = patterns with all distinct run-lengths, for all runs A351200.
%Y A351592 Cf. A000041, A008284, A018783, A047966, A181819, A182857, A238130, A305563, A319149, A351203, A351204, A351290.
%K A351592 nonn
%O A351592 0,13
%A A351592 _Gus Wiseman_, Feb 16 2022
%E A351592 More terms from _Jinyuan Wang_, Feb 14 2025