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

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, 53, 2, 96, 2, 79, 16, 21, 14, 196, 2, 23, 18, 154, 2, 166, 2, 99, 54, 27, 2, 431, 9, 85, 22, 128, 2, 303, 18, 261, 24, 33, 2, 771, 2, 35, 73, 331, 20, 422, 2, 198, 28, 216, 2, 1369

Offset: 0

Author

Gus Wiseman, Apr 22 2025

Keywords

• nonn

Examples

			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).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (2211)               (2222)
                                     (3111)               (22211)
                                     (21111)              (41111)
                                     (111111)             (221111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

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

Multiset partitions of this type are ranked by A383309.

The complement is counted by A381993, ranks A381871.

For sets we have the complement of A381994, see A381719, A382080.

Normal multiset partitions of this type are counted by A382203, sets A381718.

For distinct instead of equal block-sums we have A382427.

These partitions are ranked by A383014, nonzeros of A381995.

A000041 counts integer partitions, strict A000009.

A000688 counts factorizations into prime powers, see A381455, A381453.

A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.

A050361 counts factorizations into distinct prime powers, see A381715.

A323774 counts partitions into constant blocks with a common sum

Constant blocks with distinct sums: A381635, A381636, A381717.

Permutation with equal run-sums: A383096, A383098, A383100, A383110

Cf. A006171, A047966, A279784, A295935, A323774, A326534, A353864, A355743, A381716, A382076, A382204.

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}]

Formula

Multiset systems of this type have MM-numbers A383309 = A326534 /\ A355743.

Conjecture: We have Sum_{d|n} a(d) = A323774(n), so this is the Moebius transform of A323774.

Extensions

More terms from Jakub Buczak, May 03 2025