A383090 Number of integer partitions of n having more than one permutation with all equal run-lengths.

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 20, 28, 43, 55, 77, 107, 141, 183, 244, 312, 411, 521, 664, 837, 1069, 1328, 1667, 2069, 2578, 3166, 3929, 4791, 5895, 7168, 8749, 10594, 12883, 15500, 18741, 22493, 27069, 32334, 38760, 46133, 55065, 65367, 77686, 91905, 108927, 128431, 151674

Offset: 0

Author

Gus Wiseman, Apr 19 2025

Keywords

• nonn

Examples

			The partition (3322221) has 3 permutations with all equal run-lengths: (2323212), (2321232), (2123232), so is counted under a(15).
The partition (3322111111) has 2 permutations with all equal run-lengths: (1133112211), (1122113311), so is counted under a(16).
The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)
              (41)  (51)    (52)    (62)     (63)
                    (321)   (61)    (71)     (72)
                    (2211)  (421)   (431)    (81)
                            (3211)  (521)    (432)
                                    (3221)   (531)
                                    (3311)   (621)
                                    (4211)   (3321)
                                    (32111)  (4221)
                                             (4311)
                                             (5211)
                                             (32211)
                                             (42111)
                                             (222111)
		

Crossrefs

For no choices we have A382915, ranks A382879.

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

Partitions of this type are ranked by A383089 = positions of terms > 1 in A382857.

The complement is A383091, counted by A383092.

For a unique choice we have A383094, ranks A383112.

The complement for run-sums is A383095 + A383096, ranks A383099 \/ A383100.

For run-sums we have A383097, ranked by A383015 = positions of terms > 1 in A382877.

For distinct instead of equal run-lengths we have A383111, ranks A383113.

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

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.

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

Cf. A047993, A237984, A329739, A381541, A381871, A382076, A382771.

Programs

Mathematica

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

Formula

The complement is counted by A383094 + A382915, ranks A383112 \/ A382879.

Extensions

More terms from Bert Dobbelaere, Apr 26 2025