cp's OEIS Frontend

A332746 Number of integer partitions of n such that either the run-lengths or the negated run-lengths are unimodal.

%I A332746 #9 Feb 16 2025 08:33:59
%S A332746 1,1,2,3,5,7,11,15,22,30,42,56,77,101,134,174,227,291,373,473,598,748,
%T A332746 936,1163,1437,1771,2170,2651,3226,3916,4727,5702,6846,8205,9793,
%U A332746 11681,13866,16462,19452,22976,27041,31820,37276,43693,51023,59559,69309,80664
%N A332746 Number of integer partitions of n such that either the run-lengths or the negated run-lengths are unimodal.
%C A332746 First differs from A000041 at a(14) = 134, A000041(14) = 135.
%C A332746 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332746 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332746 The only partition not counted under a(14) = 134 is (4,3,3,2,1,1), whose run-lengths (1,2,1,2) are neither unimodal nor is their negation.
%t A332746 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332746 Table[Length[Select[IntegerPartitions[n],unimodQ[Length/@Split[#]]||unimodQ[-Length/@Split[#]]&]],{n,0,30}]
%Y A332746 Looking only at the original run-lengths gives A332281.
%Y A332746 Looking only at the negated run-lengths gives A332639.
%Y A332746 The complement is counted by A332640.
%Y A332746 The Heinz numbers of partitions not in this class are A332643.
%Y A332746 Unimodal compositions are A001523.
%Y A332746 Partitions with unimodal run-lengths are A332280.
%Y A332746 Compositions whose negation is unimodal are A332578.
%Y A332746 Partitions whose negated run-lengths are unimodal are A332638.
%Y A332746 Run-lengths are neither weakly increasing nor weakly decreasing: A332641.
%Y A332746 Run-lengths and negated run-lengths are both unimodal: A332745.
%Y A332746 Cf. A007052, A025065, A100883, A115981, A181819, A332283, A332577, A332578, A332642, A332669, A332726, A332831.
%K A332746 nonn
%O A332746 0,3
%A A332746 _Gus Wiseman_, Feb 27 2020