A383095 Number of integer partitions of n having exactly one permutation with all equal run-sums.

1, 1, 2, 2, 3, 2, 6, 2, 4, 5, 6, 2, 12, 2, 6, 8, 5, 2, 20, 2, 12, 8, 6, 2, 20, 5, 6, 12, 12, 2, 34, 2, 6, 8, 6, 8, 45, 2, 6, 8, 20, 2, 34, 2, 12, 28, 6, 2, 30, 5, 20, 8, 12, 2, 52, 8, 20, 8, 6, 2, 78, 2, 6, 28, 7, 8, 34, 2, 12, 8, 34, 2, 80, 2, 6, 28, 12, 8, 34, 2, 30, 25

Offset: 0

Author

Gus Wiseman, Apr 16 2025

Keywords

• nonn

Examples

			The partition (2,2,1,1) has permutation (2,1,1,2) so is counted under a(6).
The a(1) = 1 through a(10) = 6 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      33111      22222
                           2211             11111111  3111111    2221111
                           21111                      111111111  22111111
                           111111                                1111111111
		

Crossrefs

For distinct instead of equal run-sums we have A000005.

For run-lengths instead of sums we have A383094.

The complement is counted by A383096 + A383097, ranks A383100 \/ A383015.

These partitions are ranked by A383099 = positions of 1 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.

A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.

Cf. A329738, A353839, A353832, A353932, A354584, A381636, A381871, A382076, A382876.

Programs

Mathematica

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

Extensions

More terms from Bert Dobbelaere, Apr 26 2025