cp's OEIS Frontend

A332745 Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.

%I A332745 #7 Mar 01 2020 20:42:20
%S A332745 1,1,2,3,5,7,11,15,21,29,39,51,68,87,113,143,183,228,289,354,443,544,
%T A332745 672,812,1001,1202,1466,1758,2123,2525,3046,3606,4308,5089,6054,7102,
%U A332745 8430,9855,11621,13571,15915,18500,21673,25103,29245,33835,39296,45277,52470
%N A332745 Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.
%C A332745 Also partitions whose run-lengths and negated run-lengths are both unimodal.
%e A332745 The a(8) = 21 partitions are:
%e A332745   (8)     (44)     (2222)
%e A332745   (53)    (332)    (22211)
%e A332745   (62)    (422)    (32111)
%e A332745   (71)    (431)    (221111)
%e A332745   (521)   (3311)   (311111)
%e A332745   (611)   (4211)   (2111111)
%e A332745   (5111)  (41111)  (11111111)
%e A332745 Missing from this list is only (3221).
%t A332745 Table[Length[Select[IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,30}]
%Y A332745 The complement is counted by A332641.
%Y A332745 The Heinz numbers of partitions not in this class are A332831.
%Y A332745 The case of run-lengths of compositions is A332835.
%Y A332745 Only weakly decreasing is A100882.
%Y A332745 Only weakly increasing is A100883.
%Y A332745 Unimodal compositions are A001523.
%Y A332745 Non-unimodal compositions are A115981.
%Y A332745 Partitions with unimodal run-lengths are A332280.
%Y A332745 Partitions whose negated run-lengths are unimodal are A332638.
%Y A332745 Compositions with unimodal run-lengths are A332726.
%Y A332745 Compositions that are neither weakly increasing nor decreasing are A332834.
%Y A332745 Cf. A025065, A059204, A181819, A328509, A332281, A332283, A332577, A332578, A332640, A332727, A332742, A332833.
%K A332745 nonn
%O A332745 0,3
%A A332745 _Gus Wiseman_, Feb 29 2020