A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

1, 1, 2, 2, 4, 2, 7, 2, 7, 5, 7, 2, 19, 2, 7, 8, 14, 2, 27, 2, 24, 8, 7, 2, 58, 5, 7, 13, 30, 2, 72, 2, 38, 8, 7, 8, 135, 2, 7, 8, 91, 2, 112, 2, 45, 38, 7, 2, 258, 5, 51, 8, 54, 2, 208, 8, 143, 8, 7, 2, 525, 2, 7, 44, 153, 8, 256, 2, 75, 8, 136, 2, 891, 2, 7, 57, 87, 8

Offset: 0

Author

Gus Wiseman, Apr 17 2025

Keywords

• nonn

Examples

			The partition (4,4,4,2,2,1,1,1,1) has permutations (4,2,2,4,1,1,1,1,4) and (4,1,1,1,1,4,2,2,4) so is counted under a(20).
The a(1) = 1 through a(10) = 7 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              211          222              422       33111      22222
              1111         2211             2222      3111111    511111
                           3111             41111     111111111  2221111
                           21111            221111               22111111
                           111111           11111111             1111111111
		

Crossrefs

For distinct instead of equal run-sums we appear to have A382427.

For run-lengths instead of sums we have A383013, ranked by complement of A382879.

The case of a unique choice is A383095, ranks A383099 = positions of 1 in A382877.

The complement is counted by A383096, ranks A383100 = positions of 0 in A382877.

These partitions are ranked by A383110.

The case of more than one choice is A383097, ranks A383015.

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.

Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A382076, A382857, A382876, A383094, A383112.

Programs

Mathematica

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

Formula

a(n) = A383097(n) + A383095(n), ranks A383015 \/ A383099.

Extensions

More terms from Bert Dobbelaere, Apr 26 2025