cp's OEIS Frontend

A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.

%I A381993 #13 Mar 31 2025 21:54:22
%S A381993 0,0,0,1,1,5,4,13,13,25,33,54,54,99,124,166,207,295,352,488,591,780,
%T A381993 987,1253,1488,1951,2419,2993,3665,4563,5508,6840,8270,10127,12289,
%U A381993 14869,17781,21635,25992,31167,37184,44581,53008,63259,75076,89080,105531,124752,146842,173516,204141,239921,281461,329929,385852
%N A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.
%e A381993 The multiset partition {{2},{2},{1,1},{1,1}} has both properties (constant blocks and common sum), so (2,2,1,1,1,1) is not counted under a(8). We can also use {{2,2},{1,1,1,1}}.
%e A381993 The a(3) = 1 through a(8) = 13 partitions:
%e A381993   (21)  (31)  (32)    (42)   (43)      (53)
%e A381993               (41)    (51)   (52)      (62)
%e A381993               (221)   (321)  (61)      (71)
%e A381993               (311)   (411)  (322)     (332)
%e A381993               (2111)         (331)     (431)
%e A381993                              (421)     (521)
%e A381993                              (511)     (611)
%e A381993                              (2221)    (3221)
%e A381993                              (3211)    (3311)
%e A381993                              (4111)    (4211)
%e A381993                              (22111)   (5111)
%e A381993                              (31111)   (32111)
%e A381993                              (211111)  (311111)
%t A381993 mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
%t A381993 Table[Length[Select[IntegerPartitions[n],Length[Select[Join@@@Tuples[mce/@Split[#]],SameQ@@Total/@#&]]==0&]],{n,0,30}]
%Y A381993 Twice-partitions of this type (constant with equal) are counted by A279789.
%Y A381993 Multiset partitions of this type are ranked by A326534 /\ A355743.
%Y A381993 For distinct instead of equal block-sums we have A381717.
%Y A381993 These partitions are ranked by A381871, zeros of A381995.
%Y A381993 For strict instead of constant blocks we have A381994, see A381719, A382080.
%Y A381993 The strict case is A382076.
%Y A381993 Normal multiset partitions of this type are counted by A382204.
%Y A381993 A001055 counts factorizations, strict A045778.
%Y A381993 A050361 counts factorizations into distinct prime powers, see A381715.
%Y A381993 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A381993 Cf. A000688, A006171, A047966, A265947, A279784, A381453, A381455, A381635, A381636, A381992.
%K A381993 nonn
%O A381993 0,6
%A A381993 _Gus Wiseman_, Mar 17 2025
%E A381993 a(31)-a(54) from _Robert Price_, Mar 31 2025