cp's OEIS Frontend

A072707 Number of non-unimodal compositions of n into distinct terms.

%I A072707 #18 Aug 25 2025 02:14:28
%S A072707 0,0,0,0,0,0,2,2,4,6,24,26,46,64,100,224,276,416,590,850,1144,2214,
%T A072707 2644,3938,5282,7504,9776,13704,21984,27632,38426,51562,69844,91950,
%U A072707 123504,159658,246830,303400,416068,540480,730268,933176,1248110
%N A072707 Number of non-unimodal compositions of n into distinct terms.
%C A072707 Also the number of compositions of n into distinct terms whose negation is not unimodal. - _Gus Wiseman_, Mar 05 2020
%H A072707 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A072707 a(n) = A032020(n) - A072706(n) = Sum_{k} A059204(k) * A060016(n, k).
%e A072707 a(6)=2 since 6 can be written as 2+1+3 or 3+1+2.
%e A072707 From _Gus Wiseman_, Mar 05 2020: (Start)
%e A072707 The a(6) = 2 through a(9) = 6 strict compositions:
%e A072707   (2,1,3)  (2,1,4)  (2,1,5)  (2,1,6)
%e A072707   (3,1,2)  (4,1,2)  (3,1,4)  (3,1,5)
%e A072707                     (4,1,3)  (3,2,4)
%e A072707                     (5,1,2)  (4,2,3)
%e A072707                              (5,1,3)
%e A072707                              (6,1,2)
%e A072707 (End)
%t A072707 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A072707 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!unimodQ[#]&]],{n,0,16}] (* _Gus Wiseman_, Mar 05 2020 *)
%Y A072707 The complement is counted by A072706.
%Y A072707 The non-strict version is A115981.
%Y A072707 The case where the negation is not unimodal either is A332874.
%Y A072707 Unimodal compositions are A001523.
%Y A072707 Strict compositions are A032020.
%Y A072707 Non-unimodal permutations are A059204.
%Y A072707 A triangle for strict unimodal compositions is A072705.
%Y A072707 Non-unimodal sequences covering an initial interval are A328509.
%Y A072707 Numbers whose prime signature is not unimodal are A332282.
%Y A072707 Strict partitions whose 0-appended differences are not unimodal are A332286.
%Y A072707 Compositions whose negation is unimodal are A332578.
%Y A072707 Compositions whose negation is not unimodal are A332669.
%Y A072707 Non-unimodal compositions covering an initial interval are A332743.
%Y A072707 Cf. A007052, A072704, A107429, A227038, A329398, A332281, A332284, A332285, A332579, A332870.
%K A072707 nonn,changed
%O A072707 0,7
%A A072707 _Henry Bottomley_, Jul 04 2002