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

0,6

Links

Table of n, a(n) for n=0..54.

Example

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)

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}]

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.

Sequence in context: A147685 A353151 A078930 * A329162 A383096 A344817

Adjacent sequences: A381990 A381991 A381992 * A381994 A381995 A381996

Keyword

nonn

Author

Gus Wiseman, Mar 17 2025

Extensions

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

Status

approved