cp's OEIS Frontend

A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

%I A383093 #13 May 04 2025 10:00:57
%S A383093 1,1,2,2,4,2,7,2,9,5,9,2,23,2,11,10,24,2,33,2,36,12,15,2,87,7,17,17,
%T A383093 53,2,96,2,79,16,21,14,196,2,23,18,154,2,166,2,99,54,27,2,431,9,85,22,
%U A383093 128,2,303,18,261,24,33,2,771,2,35,73,331,20,422,2,198,28,216,2,1369
%N A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.
%F A383093 Multiset systems of this type have MM-numbers A383309 = A326534 /\ A355743.
%F A383093 Conjecture: We have Sum_{d|n} a(d) = A323774(n), so this is the Moebius transform of A323774.
%e A383093 The partition (4,4,2,2,2,2,1,1,1,1,1,1,1,1) has two partitions into constant blocks with a common sum: {{4,4},{2,2,2,2},{1,1,1,1,1,1,1,1}} and {{4},{4},{2,2},{2,2},{1,1,1,1},{1,1,1,1}}, so is counted under a(24).
%e A383093 The a(1) = 1 through a(8) = 9 partitions:
%e A383093   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A383093        (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
%e A383093                     (211)            (222)                (422)
%e A383093                     (1111)           (2211)               (2222)
%e A383093                                      (3111)               (22211)
%e A383093                                      (21111)              (41111)
%e A383093                                      (111111)             (221111)
%e A383093                                                           (2111111)
%e A383093                                                           (11111111)
%t A383093 mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
%t A383093 Table[Length[Select[IntegerPartitions[n],Length[Select[Join@@@Tuples[mce/@Split[#]],SameQ@@Total/@#&]]>0&]],{n,0,30}]
%Y A383093 Twice-partitions of this type (constant with common) are counted by A279789.
%Y A383093 Multiset partitions of this type are ranked by A383309.
%Y A383093 The complement is counted by A381993, ranks A381871.
%Y A383093 For sets we have the complement of A381994, see A381719, A382080.
%Y A383093 Normal multiset partitions of this type are counted by A382203, sets A381718.
%Y A383093 For distinct instead of equal block-sums we have A382427.
%Y A383093 These partitions are ranked by A383014, nonzeros of A381995.
%Y A383093 A000041 counts integer partitions, strict A000009.
%Y A383093 A000688 counts factorizations into prime powers, see A381455, A381453.
%Y A383093 A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
%Y A383093 A050361 counts factorizations into distinct prime powers, see A381715.
%Y A383093 A323774 counts partitions into constant blocks with a common sum
%Y A383093 Constant blocks with distinct sums: A381635, A381636, A381717.
%Y A383093 Permutation with equal run-sums: A383096, A383098, A383100, A383110
%Y A383093 Cf. A006171, A047966, A279784, A295935, A323774, A326534, A353864, A355743, A381716, A382076, A382204.
%K A383093 nonn
%O A383093 0,3
%A A383093 _Gus Wiseman_, Apr 22 2025
%E A383093 More terms from _Jakub Buczak_, May 03 2025