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

0,3

Links

Table of n, a(n) for n=0..72.

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.

Example

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)

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 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.

Sequence in context: A348219 A122977 A378596 * A275870 A321721 A359102

Adjacent sequences: A383090 A383091 A383092 * A383094 A383095 A383096

Keyword

nonn

Author

Gus Wiseman, Apr 22 2025

Extensions

More terms from Jakub Buczak, May 03 2025

Status

approved