cp's OEIS Frontend

A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

%I A351293 #19 Aug 13 2025 22:18:06
%S A351293 0,0,0,1,1,2,4,5,9,14,21,28,44,56,80,111,148,192,264,335,447,575,743,
%T A351293 937,1213,1513,1924,2396,3011,3715,4646,5687,7040,8600,10556,12804,
%U A351293 15650,18897,22930,27593,33296,39884,47921,57168,68360,81295,96807,114685
%N A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.
%C A351293 First differs from A336866 (non-Wilf partitions) at a(9) = 14, A336866(9) = 15, the difference being the partition (2,2,2,1,1,1).
%C A351293 See A239455 for the definition of Look-and-Say partitions.
%F A351293 a(n) = A000041(n) - A239455(n).
%e A351293 The a(3) = 1 through a(9) = 14 partitions:
%e A351293   (21)  (31)  (32)  (42)    (43)    (53)     (54)
%e A351293               (41)  (51)    (52)    (62)     (63)
%e A351293                     (321)   (61)    (71)     (72)
%e A351293                     (2211)  (421)   (431)    (81)
%e A351293                             (3211)  (521)    (432)
%e A351293                                     (3221)   (531)
%e A351293                                     (3311)   (621)
%e A351293                                     (4211)   (3321)
%e A351293                                     (32111)  (4221)
%e A351293                                              (4311)
%e A351293                                              (5211)
%e A351293                                              (32211)
%e A351293                                              (42111)
%e A351293                                              (321111)
%t A351293 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A351293 Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]==0&]],{n,0,15}] (* _Gus Wiseman_, Aug 13 2025 *)
%Y A351293 The complement is counted by A239455, ranked by A351294.
%Y A351293 These are all non-Wilf partitions (counted by A336866, ranked by A130092).
%Y A351293 A variant for runs is A351203, complement A351204, ranked by A351201.
%Y A351293 These partitions appear to be ranked by A351295.
%Y A351293 Non-Wilf partitions in the complement are counted by A351592.
%Y A351293 A000569 = graphical partitions, complement A339617.
%Y A351293 A032020 = number of binary expansions with all distinct run-lengths.
%Y A351293 A044813 = numbers whose binary expansion has all distinct run-lengths.
%Y A351293 A098859 = Wilf partitions (distinct multiplicities), ranked by A130091.
%Y A351293 A181819 = Heinz number of the prime signature of n (prime shadow).
%Y A351293 A329738 = compositions with all equal run-lengths.
%Y A351293 A329739 = compositions with all distinct run-lengths, for all runs A351013.
%Y A351293 A351017 = binary words with all distinct run-lengths, for all runs A351016.
%Y A351293 A351292 = patterns with all distinct run-lengths, for all runs A351200.
%Y A351293 Cf. A000041, A008284, A047966, A182857, A225485, A238130, A297770, A304660, A305563, A329740, A329746, A351202, A351291.
%K A351293 nonn
%O A351293 0,6
%A A351293 _Gus Wiseman_, Feb 16 2022
%E A351293 Edited by _Gus Wiseman_, Aug 12 2025