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

0,7

Links

Table of n, a(n) for n=0..52.

Example

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)

Mathematica

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

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.

Sequence in context: A272196 A335057 A039887 * A216162 A114215 A292302

Adjacent sequences: A382912 A382913 A382914 * A382916 A382917 A382918

Keyword

nonn

Author

Gus Wiseman, Apr 12 2025

Extensions

More terms from Bert Dobbelaere, Apr 26 2025

Status

approved