A382915 Number of integer partitions of n having no permutation with all equal run-lengths.

0, 0, 0, 0, 0, 1, 2, 4, 4, 9, 11, 18, 21, 34, 41, 55, 69, 98, 120, 160, 189, 249, 309, 396, 472, 605, 734, 913, 1099, 1371, 1632, 2021, 2406, 2937, 3514, 4251, 5039, 6101, 7221, 8646, 10205, 12209, 14347, 17086, 20041, 23713, 27807, 32803, 38262, 45043, 52477, 61471, 71496

Offset: 0

Author

Gus Wiseman, Apr 12 2025

Keywords

• nonn

Examples

			The partition y = (2,2,1,1,1) has permutations and run-lengths:
  (2,2,1,1,1) (2,3)
  (2,1,2,1,1) (1,1,1,2)
  (2,1,1,2,1) (1,2,1,1)
  (2,1,1,1,2) (1,3,1)
  (1,2,2,1,1) (1,2,2)
  (1,2,1,2,1) (1,1,1,1,1)
  (1,2,1,1,2) (1,1,2,1)
  (1,1,2,2,1) (2,2,1)
  (1,1,2,1,2) (2,1,1,1)
  (1,1,1,2,2) (3,2)
Since (1,2,1,2,1) has all equal run-lengths (1,1,1,1,1), y is not counted under a(7).
The a(5) = 1 through a(10) = 11 partitions:
  (2111)  (3111)   (2221)    (5111)     (3222)      (3331)
          (21111)  (4111)    (41111)    (6111)      (4222)
                   (31111)   (311111)   (22221)     (7111)
                   (211111)  (2111111)  (51111)     (61111)
                                        (321111)    (421111)
                                        (411111)    (511111)
                                        (2211111)   (3211111)
                                        (3111111)   (4111111)
                                        (21111111)  (22111111)
                                                    (31111111)
                                                    (211111111)
		

Crossrefs

The complement for distinct run-lengths is A239455, ranked by A351294.

For distinct instead of equal run-lengths we have A351293, ranked by A351295.

These partitions are ranked by A382879, by signature A382914.

The complement is counted by A383013.

A000041 counts integer partitions, strict A000009.

A056239 adds up prime indices, row sums of A112798.

A304442 counts partitions with equal run-sums, ranks A353833.

A329738 counts compositions with equal run-lengths, ranks A353744.

A382857 counts permutations of prime indices with equal run-lengths.

Cf. A003242, A047966, A238279, A329739, A351201, A351290, A351596, A382773.

Programs

Mathematica

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

Extensions

More terms from Bert Dobbelaere, Apr 26 2025