cp's OEIS Frontend

A383710 Number of integer partitions of n such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

%I A383710 #7 May 08 2025 22:57:09
%S A383710 0,0,1,1,3,4,6,10,15,22,29,42,59,79,108,140,190,247,324,417,541
%N A383710 Number of integer partitions of n such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.
%C A383710 Also the number of integer partitions of n whose normal multiset (in which i appears y_i times) is not a Look-and-Say partition.
%e A383710 For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is not counted under a(6).
%e A383710 The a(2) = 1 through a(8) = 15 partitions:
%e A383710   (11)  (111)  (22)    (221)    (222)     (322)      (332)
%e A383710                (211)   (311)    (411)     (331)      (422)
%e A383710                (1111)  (2111)   (2211)    (511)      (611)
%e A383710                        (11111)  (3111)    (2221)     (2222)
%e A383710                                 (21111)   (3211)     (3221)
%e A383710                                 (111111)  (4111)     (3311)
%e A383710                                           (22111)    (4211)
%e A383710                                           (31111)    (5111)
%e A383710                                           (211111)   (22211)
%e A383710                                           (1111111)  (32111)
%e A383710                                                      (41111)
%e A383710                                                      (221111)
%e A383710                                                      (311111)
%e A383710                                                      (2111111)
%e A383710                                                      (11111111)
%t A383710 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
%t A383710 Table[Length[Select[IntegerPartitions[n], pof[#]=={}&]], {n,0,15}]
%Y A383710 These partitions have Heinz numbers A382912.
%Y A383710 The number of such families for each Heinz number is A383706.
%Y A383710 The complement is counted by A383708, ranks A382913.
%Y A383710 Without ones we have A383711, complement A383533.
%Y A383710 A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
%Y A383710 A098859 counts partitions with distinct multiplicities, compositions A242882.
%Y A383710 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A383710 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A383710 Cf. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381454, A383013.
%K A383710 nonn,more
%O A383710 0,5
%A A383710 _Gus Wiseman_, May 07 2025