cp's OEIS Frontend

A383533 Number of integer partitions of n with no ones such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

%I A383533 #9 May 08 2025 22:57:03
%S A383533 1,0,1,1,1,2,3,3,4,5,8,8,11,13,17,22,25,30,37,44,53,69,77,93,111,130,
%T A383533 153,181,220,249,295
%N A383533 Number of integer partitions of n with no ones such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.
%C A383533 The Heinz numbers of these partitions are the odd terms of A382913.
%C A383533 Also the number of integer partitions y of n with no ones such that the normal multiset (in which i appears y_i times) is a Look-and-Say partition.
%e A383533 For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is counted under a(6).
%e A383533 The a(2) = 1 through a(10) = 8 partitions:
%e A383533   (2)  (3)  (4)  (5)    (6)    (7)    (8)    (9)      (10)
%e A383533                  (3,2)  (3,3)  (4,3)  (4,4)  (5,4)    (5,5)
%e A383533                         (4,2)  (5,2)  (5,3)  (6,3)    (6,4)
%e A383533                                       (6,2)  (7,2)    (7,3)
%e A383533                                              (4,3,2)  (8,2)
%e A383533                                                       (4,3,3)
%e A383533                                                       (4,4,2)
%e A383533                                                       (5,3,2)
%t A383533 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
%t A383533 Table[Length[Select[IntegerPartitions[n], FreeQ[#,1]&&!pof[#]=={}&]],{n,0,15}]
%Y A383533 The number of such families is A383706.
%Y A383533 Allowing ones gives A383708 (ranks A382913), complement A383710 (ranks A382912).
%Y A383533 The complement is counted by A383711.
%Y A383533 A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
%Y A383533 A098859 counts partitions with distinct multiplicities, compositions A242882.
%Y A383533 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A383533 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A383533 Cf. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381441, A381454, A383013.
%K A383533 nonn,more
%O A383533 0,6
%A A383533 _Gus Wiseman_, May 07 2025