A383092 Number of integer partitions of n having at most one permutation with all equal run-lengths.

1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 22, 28, 34, 46, 58, 69, 90, 114, 141, 178, 216, 271, 338, 418, 506, 630, 769, 941, 1140, 1399, 1675, 2051, 2454, 2975, 3561, 4289, 5094, 6137, 7274, 8692, 10269, 12249, 14414, 17128, 20110, 23767, 27872, 32849, 38346, 45094, 52552, 61533

Offset: 0

Author

Gus Wiseman, Apr 19 2025

Keywords

• nonn

Examples

			The partition (222211) has 1 permutation with all equal run-lengths: (221122), so is counted under a(10).
The partition (33211111) has no permutation with all equal run-lengths, so is counted under a(13).
The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
		

Crossrefs

For no choices we have A382915, ranks A382879.

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

The complement is A383090, ranks A383089.

Partitions of this type are ranked by A383091 = positions of terms <= 1 in A382857.

For a unique choice we have A383094, ranks A383112.

For run-sums instead of lengths we have A383095 + A383096, ranks A383099 \/ A383100.

The complement for run-sums is A383097, ranks A383015, positions of terms > 1 in A382877.

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. A006171, A047993, A362608, A381871, A382076, A382771, A383111.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Length/@Split[#]&]]<=1&]],{n,0,15}]

Formula

a(n) = A382915(n) + A383094(n).

Extensions

More terms from Bert Dobbelaere, Apr 26 2025