cp's OEIS Frontend

A383708 Number of integer partitions of n such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

%I A383708 #10 May 08 2025 22:55:53
%S A383708 1,1,2,2,3,5,5,7,8,13,14,18,22,27,36,41,50,61,73,86
%N A383708 Number of integer partitions of n such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.
%C A383708 Also the number of integer partitions y of n whose normal multiset (in which i appears y_i times) is a Look-and-Say partition.
%e A383708 For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is counted under a(6).
%e A383708 The a(1) = 1 through a(9) = 8 partitions:
%e A383708   (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
%e A383708             (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
%e A383708                           (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
%e A383708                                  (5,1)    (6,1)    (6,2)    (7,2)
%e A383708                                  (3,2,1)  (4,2,1)  (7,1)    (8,1)
%e A383708                                                    (4,3,1)  (4,3,2)
%e A383708                                                    (5,2,1)  (5,3,1)
%e A383708                                                             (6,2,1)
%t A383708 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
%t A383708 Table[Length[Select[IntegerPartitions[n], pof[#]!={}&]],{n,15}]
%Y A383708 These partitions have Heinz numbers A382913.
%Y A383708 Without ones we have A383533, complement A383711.
%Y A383708 The number of such families for each Heinz number is A383706.
%Y A383708 The complement is counted by A383710, ranks A382912.
%Y A383708 A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
%Y A383708 A098859 counts partitions with distinct multiplicities, compositions A242882.
%Y A383708 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A383708 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A383708 Cf. A044813, A047966, A089259, A116540, A091602, A130091, A317141, A351013, A381441, A382771, A383013.
%K A383708 nonn,more
%O A383708 0,3
%A A383708 _Gus Wiseman_, May 07 2025