This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A332640 #10 Feb 16 2025 08:33:59
%S A332640 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,6,12,17,29,44,66,92,138,187,266,
%T A332640 359,492,649,877,1140,1503,1938,2517,3202,4111,5175,6563,8209,10297,
%U A332640 12763,15898,19568,24152,29575,36249,44090,53737,65022,78752,94873,114294
%N A332640 Number of integer partitions of n such that neither the run-lengths nor the negated run-lengths are unimodal.
%C A332640 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332640 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332640 The a(14) = 1 through a(18) = 12 partitions:
%e A332640 (433211) (533211) (443221) (544211) (544311)
%e A332640 (4332111) (633211) (733211) (553221)
%e A332640 (5332111) (4333211) (644211)
%e A332640 (43321111) (6332111) (833211)
%e A332640 (53321111) (4432221)
%e A332640 (433211111) (5333211)
%e A332640 (5442111)
%e A332640 (7332111)
%e A332640 (43332111)
%e A332640 (63321111)
%e A332640 (533211111)
%e A332640 (4332111111)
%e A332640 For example, the partition (4,3,3,2,1,1) has run-lengths (1,2,1,2), so is counted under a(14).
%t A332640 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332640 Table[Length[Select[IntegerPartitions[n],!unimodQ[Length/@Split[#]]&&!unimodQ[-Length/@Split[#]]&]],{n,0,30}]
%Y A332640 Looking only at the original run-lengths gives A332281.
%Y A332640 Looking only at the negated run-lengths gives A332639.
%Y A332640 The Heinz numbers of these partitions are A332643.
%Y A332640 The complement is counted by A332746.
%Y A332640 Unimodal compositions are A001523.
%Y A332640 Non-unimodal permutations are A059204.
%Y A332640 Non-unimodal compositions are A115981.
%Y A332640 Partitions with unimodal run-lengths are A332280.
%Y A332640 Partitions whose negated run-lengths are unimodal are A332638.
%Y A332640 Run-lengths and negated run-lengths are not both unimodal: A332641.
%Y A332640 Compositions whose negation is not unimodal are A332669.
%Y A332640 Run-lengths and negated run-lengths are both unimodal: A332745.
%Y A332640 Cf. A007052, A025065, A100883, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332642, A332726, A332727.
%K A332640 nonn
%O A332640 0,16
%A A332640 _Gus Wiseman_, Feb 25 2020