A333190 - OEIS

A333190

Number of integer partitions of n whose run-lengths are either strictly increasing or strictly decreasing.

%I #9 May 18 2020 06:38:01

%S 1,1,2,2,4,5,7,10,13,15,21,26,29,39,49,50,68,80,92,109,129,142,181,

%T 201,227,262,317,343,404,456,516,589,677,742,870,949,1077,1207,1385,

%U 1510,1704,1895,2123,2352,2649,2877,3261,3571,3966,4363,4873,5300,5914,6466

%N Number of integer partitions of n whose run-lengths are either strictly increasing or strictly decreasing.

%e The a(1) = 1 through a(8) = 13 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (111) (22) (221) (33) (322) (44)

%e (211) (311) (222) (331) (332)

%e (1111) (2111) (411) (511) (422)

%e (11111) (3111) (2221) (611)

%e (21111) (4111) (2222)

%e (111111) (22111) (5111)

%e (31111) (22211)

%e (211111) (41111)

%e (1111111) (221111)

%e (311111)

%e (2111111)

%e (11111111)

%t Table[Length[Select[IntegerPartitions[n],Or[Less@@Length/@Split[#],Greater@@Length/@Split[#]]&]],{n,0,30}]

%Y The non-strict version is A332745.

%Y The generalization to compositions is A333191.

%Y Partitions with distinct run-lengths are A098859.

%Y Partitions with strictly increasing run-lengths are A100471.

%Y Partitions with strictly decreasing run-lengths are A100881.

%Y Partitions with weakly decreasing run-lengths are A100882.

%Y Partitions with weakly increasing run-lengths are A100883.

%Y Partitions with unimodal run-lengths are A332280.

%Y Partitions whose run-lengths are not increasing nor decreasing are A332641.

%Y Compositions whose run-lengths are unimodal or co-unimodal are A332746.

%Y Compositions that are neither increasing nor decreasing are A332834.

%Y Strictly increasing or strictly decreasing compositions are A333147.

%Y Compositions with strictly increasing run-lengths are A333192.

%Y Numbers with strictly increasing prime multiplicities are A334965.

%Y Cf. A032020, A059204, A072706, A332726, A332831, A332833, A332835, A333149.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 17 2020