A383094 Number of integer partitions of n having exactly one permutation with all equal run-lengths.

1, 1, 2, 2, 4, 4, 5, 6, 9, 7, 11, 10, 13, 12, 17, 14, 21, 16, 21, 18, 27, 22, 29, 22, 34, 25, 35, 28, 41, 28, 43, 30, 48, 38, 47, 38, 55, 36, 53, 46, 64, 40, 67, 42, 69, 54, 65, 46, 84, 51, 75, 62, 83, 52, 86, 62, 94, 70, 83, 58, 111, 60, 89, 80, 106, 74, 115, 66, 111

Offset: 0

Author

Gus Wiseman, Apr 20 2025

Keywords

• nonn

Examples

			The partition (222211) has exactly one permutation with all equal run-lengths: (221122), so is counted under a(10).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (11111)  (411)     (511)      (422)
                                     (111111)  (22111)    (611)
                                               (1111111)  (2222)
                                                          (22211)
                                                          (221111)
                                                          (11111111)
		

Crossrefs

The complement is ranked by A382879 \/ A383089.

For no choices we have A382915, ranks A382879.

For at least one choice we have A383013, for run-sums A383098, ranks A383110.

For more than one choice we have A383090, ranks A383089.

For at most one choice we have A383092, ranks A383091.

For run-sums instead of lengths we have A383095, ranks A383099.

Partitions of this type are ranked by A383112 = positions of 1 in A382857.

Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.

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

A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.

A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.

Cf. A047993, A362608, A381871, A382076, A382771, A383096, A383097, A383111.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]==1&]],{n,0,20}]

Extensions

More terms from Bert Dobbelaere, Apr 26 2025