A357878 Number of integer partitions of n whose run-sums are not weakly decreasing.

0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 40, 55, 79, 104, 150, 196, 270, 350, 467, 600, 786, 997, 1293, 1632, 2077, 2597, 3283, 4067, 5088, 6268, 7769, 9517, 11704, 14238, 17405, 21092, 25598, 30861, 37278, 44729, 53742, 64226, 76811, 91448, 108929, 129174

Offset

0,8

Comments

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

Links

Table of n, a(n) for n=0..48.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Example

The a(0) = 0 through a(9) = 8 partitions:
  .  .  .  .  .  (2111)  (21111)  (322)     (3221)     (3222)
                                  (31111)   (32111)    (32211)
                                  (211111)  (311111)   (42111)
                                            (2111111)  (321111)
                                                       (411111)
                                                       (2211111)
                                                       (3111111)
                                                       (21111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], !LessEqual@@Total/@Split[Reverse[#]]&]], {n, 0, 30}]

Crossrefs

The complement is counted by A304405, ranked by A357875.

Number of rows in A354584 summing to n that are weakly increasing.

The opposite (not weakly increasing) version is A357865, ranked by A357850.

These partitions are ranked by A357876.

A000041 counts integer partitions, strict A000009.

A304442 counts partitions with equal run-sums, distinct A353837.

Cf. A047966, A098859, A239312, A275870, A304406, A304428, A304430, A353832, A353864, A353932.

Sequence in context: A084421 A271723 A212545 * A358910 A212546 A212547

Adjacent sequences: A357875 A357876 A357877 * A357879 A357880 A357881

Keyword

nonn

Author

Gus Wiseman, Oct 18 2022

Status

approved