A072707 - OEIS

%I #13 Oct 21 2024 20:24:58

%S 0,0,0,0,0,0,2,2,4,6,24,26,46,64,100,224,276,416,590,850,1144,2214,

%T 2644,3938,5282,7504,9776,13704,21984,27632,38426,51562,69844,91950,

%U 123504,159658,246830,303400,416068,540480,730268,933176,1248110

%N Number of non-unimodal compositions of n into distinct terms.

%C Also the number of compositions of n into distinct terms whose negation is not unimodal. - Gus Wiseman, Mar 05 2020

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>

%F a(n) = A032020(n) - A072706(n) = Sum_{k} A059204(k) * A060016(n, k).

%e a(6)=2 since 6 can be written as 2+1+3 or 3+1+2.

%e From _Gus Wiseman_, Mar 05 2020: (Start)

%e The a(6) = 2 through a(9) = 6 strict compositions:

%e   (2,1,3)  (2,1,4)  (2,1,5)  (2,1,6)

%e   (3,1,2)  (4,1,2)  (3,1,4)  (3,1,5)

%e                     (4,1,3)  (3,2,4)

%e                     (5,1,2)  (4,2,3)

%e                              (5,1,3)

%e                              (6,1,2)

%e (End)

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!unimodQ[#]&]],{n,0,16}] (* _Gus Wiseman_, Mar 05 2020 *)

%Y The complement is counted by A072706.

%Y The non-strict version is A115981.

%Y The case where the negation is not unimodal either is A332874.

%Y Unimodal compositions are A001523.

%Y Strict compositions are A032020.

%Y Non-unimodal permutations are A059204.

%Y A triangle for strict unimodal compositions is A072705.

%Y Non-unimodal sequences covering an initial interval are A328509.

%Y Numbers whose prime signature is not unimodal are A332282.

%Y Strict partitions whose 0-appended differences are not unimodal are A332286.

%Y Compositions whose negation is unimodal are A332578.

%Y Compositions whose negation is not unimodal are A332669.

%Y Non-unimodal compositions covering an initial interval are A332743.

%Y Cf. A007052, A072704, A107429, A227038, A329398, A332281, A332284, A332285, A332579, A332870.

%K nonn

%O 0,7

%A _Henry Bottomley_, Jul 04 2002