A383096 Number of integer partitions of n having no permutation with all equal run-sums.

0, 0, 0, 1, 1, 5, 4, 13, 15, 25, 35, 54, 58, 99, 128, 168, 217, 295, 358, 488, 603, 784, 995, 1253, 1517, 1953, 2429, 2997, 3688, 4563, 5532, 6840, 8311, 10135, 12303, 14875, 17842, 21635, 26008, 31177, 37247, 44581, 53062, 63259, 75130, 89096, 105551, 124752, 147015, 173520

Offset: 0

Author

Gus Wiseman, Apr 17 2025

Keywords

• nonn

Examples

			The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (22211)
                             (211111)  (32111)
                                       (311111)
                                       (2111111)
		

Crossrefs

For distinct instead of equal run-sums we appear to have A381717, q.v.

For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.

For more than one permutation we have A383097, ranks A383015.

The complement is counted by A383098, ranks A383110

These partitions are ranked by A383100, positions of 0 in A382877.

Counting and ranking partitions by run-lengths and run-sums:

- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)

- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)

- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)

- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)

- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)

- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)

A275870 counts collapsible partitions, ranks A300273.

A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.

A353851 counts compositions with all equal run-sums, ranks A353848.

A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.

A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.

Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.

Programs

Mathematica

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

Extensions

More terms from Bert Dobbelaere, Apr 26 2025