A332577 Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 25, 30, 36, 40, 45, 54, 59, 68, 79, 86, 96, 112, 121, 135, 155, 168, 188, 214, 230, 253, 284, 308, 337, 380, 407, 445, 497, 533, 580, 645, 689, 748, 828, 885, 956, 1053, 1124, 1212, 1330, 1415, 1519, 1665, 1771

Offset: 0

Author

Gus Wiseman, Feb 24 2020

Keywords

• nonn

Comments

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Examples

			The a(1) = 1 through a(9) = 8 partitions:
  1  11  21   211   221    321     2221     3221      3321
         111  1111  2111   2211    3211     22211     22221
                    11111  21111   22111    32111     32211
                           111111  211111   221111    222111
                                   1111111  2111111   321111
                                            11111111  2211111
                                                      21111111
                                                      111111111
		

Links

• MathWorld, Unimodal Sequence

Crossrefs

Not requiring unimodality gives A000009.

A version for compositions is A227038.

Not requiring the partition to cover an initial interval gives A332280.

The complement is counted by A332579.

Unimodal compositions are A001523.

Cf. A007052, A011782, A025065, A100883, A107429, A115981, A332281, A332283, A332638, A332639, A332728.

Programs

Mathematica

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n],normQ[#]&&unimodQ[Length/@Split[#]]&]],{n,0,30}]