cp's OEIS Frontend

A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.

%I A387178 #9 Sep 03 2025 17:26:27
%S A387178 1,1,1,2,2,3,5,6,8,10,13,17,21,27,34,42,53,65,80,98,119,146,177,213,
%T A387178 258,309,370,443,528,628,745,882,1043,1229,1447,1700,1993,2333,2727,
%U A387178 3182,3707,4311,5008,5808,6727,7782,8990,10371,11952,13756,15815,18161
%N A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.
%C A387178 First differs from A052337 in having 745 instead of 746.
%C A387178 We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.
%C A387178 a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct strict integer partitions of each part.
%C A387178 Also the number of integer partitions of n with no part k whose multiplicity exceeds A000009(k).
%e A387178 The partition y = (3,3,2) has sets of strict integer partitions ({(2,1),(3)},{(2,1),(3)},{(2)}), and we have the choice ((2,1),(3),(2)) or ((3),(2,1),(2)), so y is counted under a(8).
%e A387178 The a(1) = 1 through a(9) = 10 partitions:
%e A387178   (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
%e A387178             (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
%e A387178                           (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
%e A387178                                  (5,1)    (6,1)    (6,2)    (7,2)
%e A387178                                  (3,2,1)  (3,3,1)  (7,1)    (8,1)
%e A387178                                           (4,2,1)  (3,3,2)  (4,3,2)
%e A387178                                                    (4,3,1)  (4,4,1)
%e A387178                                                    (5,2,1)  (5,3,1)
%e A387178                                                             (6,2,1)
%e A387178                                                             (3,3,2,1)
%t A387178 strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
%t A387178 Table[Length[Select[IntegerPartitions[n],Select[Tuples[strptns/@#],UnsameQ@@#&]!={}&]],{n,0,15}]
%Y A387178 For initial intervals instead of strict partitions we have A238873, ranks A387112.
%Y A387178 For divisors instead of strict partitions we have A239312, ranks A368110.
%Y A387178 The complement for divisors is A370320, ranks A355740.
%Y A387178 For prime factors instead of strict partitions we have A370592, ranks A368100.
%Y A387178 The complement for prime factors is A370593, ranks A355529.
%Y A387178 The complement for initial intervals is A387118, ranks A387113.
%Y A387178 The complement for all partitions is A387134, ranks A387577.
%Y A387178 The complement is counted by A387137, ranks A387176.
%Y A387178 These partitions are ranked by A387177.
%Y A387178 For all partitions instead of just strict partitions we have A387328, ranks A387576.
%Y A387178 The complement for constant partitions is A387329, ranks A387180.
%Y A387178 For constant partitions instead of strict partitions we have A387330, ranks A387181.
%Y A387178 A000041 counts integer partitions, strict A000009.
%Y A387178 A358914 counts twice-partitions into distinct strict partitions.
%Y A387178 A367902 counts choosable set-systems, complement A367903.
%Y A387178 Cf. A005703, A052335, A261049, A270995, A276078, A335448, A367867, A367901, A387115.
%K A387178 nonn,new
%O A387178 0,4
%A A387178 _Gus Wiseman_, Sep 02 2025