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

0, 0, 0, 1, 1, 5, 4, 13, 13, 25, 33, 54, 54, 99, 124, 166, 207, 295, 352, 488, 591, 780, 987, 1253, 1488, 1951, 2419, 2993, 3665, 4563, 5508, 6840, 8270, 10127, 12289, 14869, 17781, 21635, 25992, 31167, 37184, 44581, 53008, 63259, 75076, 89080, 105531, 124752, 146842, 173516, 204141, 239921, 281461, 329929, 385852

Offset: 0

Author

Gus Wiseman, Mar 17 2025

Keywords

• nonn

Examples

			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}}.
The a(3) = 1 through a(8) = 13 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (32111)
                             (211111)  (311111)
		

Crossrefs

Twice-partitions of this type (constant with equal) are counted by A279789.

Multiset partitions of this type are ranked by A326534 /\ A355743.

For distinct instead of equal block-sums we have A381717.

These partitions are ranked by A381871, zeros of A381995.

For strict instead of constant blocks we have A381994, see A381719, A382080.

The strict case is A382076.

Normal multiset partitions of this type are counted by A382204.

A001055 counts factorizations, strict A045778.

A050361 counts factorizations into distinct prime powers, see A381715.

A317141 counts coarsenings of prime indices, refinements A300383.

Cf. A000688, A006171, A047966, A265947, A279784, A381453, A381455, A381635, A381636, A381992.

Programs

Mathematica

mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Length[Select[Join@@@Tuples[mce/@Split[#]],SameQ@@Total/@#&]]==0&]],{n,0,30}]

Extensions

a(31)-a(54) from Robert Price, Mar 31 2025