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

0,3

Links

Table of n, a(n) for n=0..77.

Formula

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

Example

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

Mathematica

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

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.

Sequence in context: A331121 A057767 A207329 * A348219 A122977 A378596

Adjacent sequences: A383095 A383096 A383097 * A383099 A383100 A383101

Keyword

nonn

Author

Gus Wiseman, Apr 17 2025

Extensions

More terms from Bert Dobbelaere, Apr 26 2025

Status

approved