A382076 Number of integer partitions of n whose run-sums are not all equal.

0, 0, 0, 1, 1, 5, 6, 13, 15, 27, 37, 54, 64, 99, 130, 172, 220, 295, 372, 488, 615, 788, 997, 1253, 1547, 1955, 2431, 3005, 3706, 4563, 5586, 6840, 8332, 10139, 12305, 14879, 17933, 21635, 26010, 31181, 37314, 44581, 53156, 63259, 75163, 89124, 105553, 124752, 147210

Offset: 0

Author

Gus Wiseman, Apr 02 2025

Keywords

• nonn

Comments

Also the number of integer partitions of n that cannot be partitioned into distinct constant multisets with a common sum. Multiset partitions of this type are ranked by A005117 /\ A326534 /\ A355743, while twice-partitions are counted by A382524, strict case of A279789.

Examples

			The partition (3,2,1,1,1) has runs ((3),(2),(1,1,1)) with sums (3,2,3) so is counted under a(8).
The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)     (43)      (53)
              (41)    (51)     (52)      (62)
              (221)   (321)    (61)      (71)
              (311)   (411)    (322)     (332)
              (2111)  (2211)   (331)     (431)
                      (21111)  (421)     (521)
                               (511)     (611)
                               (2221)    (3221)
                               (3211)    (3311)
                               (4111)    (4211)
                               (22111)   (5111)
                               (31111)   (22211)
                               (211111)  (32111)
                                         (311111)
                                         (2111111)
		

Crossrefs

The complement is counted by A304442, ranks A353833.

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

This is the strict case of A381993, see A381995, zeros A381871.

A050361 counts factorizations into distinct prime powers, see A381715.

A304405 counts partitions with weakly decreasing run-sums, ranks A357875.

A304406 counts partitions with weakly increasing run-sums, ranks A357861.

A304428 counts partitions with strictly decreasing run-sums, ranks A357862.

A304430 counts partitions with strictly increasing run-sums, ranks A357864.

A317141 counts coarsenings of prime indices, refinements A300383.

A326534 ranks multiset partitions with a common sum.

A353837 counts partitions with distinct run-sums.

A354584 lists run-sums of weakly increasing prime indices.

A355743 ranks multiset partitions into constant blocks.

Cf. A000688, A005117, A006171, A047966, A279784, A381453, A381455, A381635, A381636, A381994, A382204.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!SameQ@@Total/@Split[#]&]],{n,0,15}]

Extensions

More terms from Bert Dobbelaere, Apr 26 2025