A383013 Number of integer partitions of n having a permutation with all equal run-lengths.

1, 1, 2, 3, 5, 6, 9, 11, 18, 21, 31, 38, 56, 67, 94, 121, 162, 199, 265, 330, 438, 543, 693, 859, 1103, 1353, 1702, 2097, 2619, 3194, 3972, 4821, 5943, 7206, 8796, 10632, 12938, 15536, 18794, 22539, 27133, 32374, 38827, 46175, 55134, 65421, 77751, 91951, 109011, 128482

Offset: 0

Author

Gus Wiseman, Apr 12 2025

Keywords

• nonn

Comments

A partition of n counts towards a(n) if and only if #p + g >= 2*L where #p is the number of parts counted with multiplicity of the partition, g is the gcd of all the frequencies of every distinct part and L is the largest frequency of a part. - David A. Corneth, Apr 27 2025

Examples

			The partition (2,2,1,1,1,1) has permutation (1,1,2,2,1,1) with equal run-lengths (2,2,2) so is counted under a(8).
The a(1) = 1 through a(8) = 18 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (311)    (222)     (322)      (71)
                            (11111)  (321)     (331)      (332)
                                     (411)     (421)      (422)
                                     (2211)    (511)      (431)
                                     (111111)  (3211)     (521)
                                               (22111)    (611)
                                               (1111111)  (2222)
                                                          (3221)
                                                          (3311)
                                                          (4211)
                                                          (22211)
                                                          (32111)
                                                          (221111)
                                                          (11111111)
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..82

Crossrefs

For distinct instead of equal run-lengths we have A239455, ranked by A351294.

The complement for distinct run-lengths is A351293, ranked by A351295.

The complement is counted by A382915, ranked by A382879, by signature A382914.

A000041 counts integer partitions, strict A000009.

A304442 counts partitions with equal run-sums, ranks A353833.

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

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

A382857 counts permutations of prime indices with equal run-lengths, firsts A382878.

Cf. A003242, A047966, A164707, A238279, A351201, A351290, A351291, A353837, A353851, A382858.

Programs

Mathematica

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

Extensions

More terms from Bert Dobbelaere, Apr 26 2025