cp's OEIS Frontend

A332641 Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

%I A332641 #9 Feb 16 2025 08:33:59
%S A332641 0,0,0,0,0,0,0,0,1,1,3,5,9,14,22,33,48,69,96,136,184,248,330,443,574,
%T A332641 756,970,1252,1595,2040,2558,3236,4041,5054,6256,7781,9547,11782,
%U A332641 14394,17614,21423,26083,31501,38158,45930,55299,66262,79477,94803,113214
%N A332641 Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.
%C A332641 Also partitions whose run-lengths and negated run-lengths are not both unimodal. A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332641 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332641 The a(8) = 1 through a(13) = 14 partitions:
%e A332641   (3221)  (4221)  (5221)   (4331)    (4332)     (5332)
%e A332641                   (32221)  (6221)    (5331)     (6331)
%e A332641                   (33211)  (42221)   (7221)     (8221)
%e A332641                            (322211)  (43221)    (43321)
%e A332641                            (332111)  (44211)    (44311)
%e A332641                                      (52221)    (53221)
%e A332641                                      (322221)   (62221)
%e A332641                                      (422211)   (332221)
%e A332641                                      (3321111)  (333211)
%e A332641                                                 (422221)
%e A332641                                                 (442111)
%e A332641                                                 (522211)
%e A332641                                                 (3222211)
%e A332641                                                 (33211111)
%t A332641 Table[Length[Select[IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,30}]
%Y A332641 The complement is counted by A332745.
%Y A332641 The Heinz numbers of these partitions are A332831.
%Y A332641 The case of run-lengths of compositions is A332833.
%Y A332641 Partitions whose run-lengths are weakly increasing are A100883.
%Y A332641 Partitions whose run-lengths are weakly decreasing are A100882.
%Y A332641 Partitions whose run-lengths are not unimodal are A332281.
%Y A332641 Partitions whose negated run-lengths are not unimodal are A332639.
%Y A332641 Unimodal compositions are A001523.
%Y A332641 Non-unimodal permutations are A059204.
%Y A332641 Non-unimodal compositions are A115981.
%Y A332641 Partitions with unimodal run-lengths are A332280.
%Y A332641 Partitions whose negated run-lengths are unimodal are A332638.
%Y A332641 Compositions whose negation is not unimodal are A332669.
%Y A332641 The case of run-lengths of compositions is A332833.
%Y A332641 Compositions that are neither increasing nor decreasing are A332834.
%Y A332641 Cf. A025065, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332640, A332642, A332726, A332727, A332742, A332835.
%K A332641 nonn
%O A332641 0,11
%A A332641 _Gus Wiseman_, Feb 26 2020