A383111 Number of integer partitions of n having more than one permutation with all distinct run-lengths.

0, 0, 0, 0, 1, 3, 3, 8, 9, 13, 17, 26, 27, 43, 51, 61, 78, 103, 115, 153, 174, 213, 255, 316, 354, 442, 508, 610, 701, 848, 950, 1153, 1303, 1539, 1750, 2075, 2318, 2738, 3081

Offset: 0

Author

Gus Wiseman, Apr 20 2025

Keywords

• nonn
• hard
• more

Examples

			The partition (2,1,1) has two permutations with all distinct run-lengths: (1,1,2), (2,1,1), so it is counted under a(4).
The a(4) = 1 through a(9) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (5111)     (3222)
                          (4111)    (22211)    (6111)
                          (22111)   (41111)    (22221)
                          (31111)   (221111)   (33111)
                          (211111)  (311111)   (51111)
                                    (2111111)  (222111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
		

Crossrefs

For a unique choice we have A000005, ranks A000961.

For at least one choice we have A239455, ranks A351294, conjugate A381432.

For no choices we have A351293, ranks A351295, conjugate A381433.

The complement is A351293 + A000005, ranks too dense.

For equal instead of distinct run-lengths we have A383090, ranks A383089.

These partitions are ranked by A383113 = positions of terms > 1 in A382771.

Cf. A047966, A098859, A130091, A353837, A100471, A100881, A100882, A100883

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

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

Cf. A047993, A329739, A381541, A381636, A381717, A382857, A382915, A383013, A383092, A383094, A383097.

Programs

Mathematica

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

Extensions

a(21)-a(38) from Jakub Buczak, May 04 2025