cp's OEIS Frontend

A335375 Numbers k such that the k-th composition in standard order (A066099) is neither unimodal nor co-unimodal.

%I A335375 #5 Jun 05 2020 09:57:09
%S A335375 45,54,77,89,91,93,102,108,109,110,118,141,153,155,157,166,173,177,
%T A335375 178,179,181,182,183,185,187,189,198,204,205,206,214,216,217,218,219,
%U A335375 220,221,222,230,236,237,238,246,269,281,283,285,297,301,305,306,307,309
%N A335375 Numbers k such that the k-th composition in standard order (A066099) is neither unimodal nor co-unimodal.
%C A335375 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. It is co-unimodal if its negation is unimodal.
%C A335375 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%e A335375 The sequence together with the corresponding compositions begins:
%e A335375    45: (2,1,2,1)
%e A335375    54: (1,2,1,2)
%e A335375    77: (3,1,2,1)
%e A335375    89: (2,1,3,1)
%e A335375    91: (2,1,2,1,1)
%e A335375    93: (2,1,1,2,1)
%e A335375   102: (1,3,1,2)
%e A335375   108: (1,2,1,3)
%e A335375   109: (1,2,1,2,1)
%e A335375   110: (1,2,1,1,2)
%e A335375   118: (1,1,2,1,2)
%e A335375   141: (4,1,2,1)
%e A335375   153: (3,1,3,1)
%e A335375   155: (3,1,2,1,1)
%e A335375   157: (3,1,1,2,1)
%e A335375   166: (2,3,1,2)
%e A335375   173: (2,2,1,2,1)
%e A335375   177: (2,1,4,1)
%e A335375   178: (2,1,3,2)
%e A335375   179: (2,1,3,1,1)
%t A335375 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A335375 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A335375 Select[Range[0,100],!unimodQ[stc[#]]&&!unimodQ[-stc[#]]&]
%Y A335375 Non-unimodal compositions are ranked by A335373.
%Y A335375 Non-co-unimodal compositions are ranked by A335374.
%Y A335375 Unimodal compositions are A001523.
%Y A335375 Unimodal normal sequences are A007052.
%Y A335375 Unimodal permutations are A011782.
%Y A335375 Non-unimodal permutations are A059204.
%Y A335375 Non-unimodal compositions are A115981.
%Y A335375 Non-unimodal normal sequences are A328509.
%Y A335375 Numbers with non-unimodal unsorted prime signature are A332282.
%Y A335375 Co-unimodal compositions are A332578.
%Y A335375 Numbers with non-co-unimodal unsorted prime signature are A332642.
%Y A335375 Non-co-unimodal compositions are A332669.
%Y A335375 Cf. A000120, A029931, A048793, A066099, A070939, A334299.
%Y A335375 Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.
%K A335375 nonn
%O A335375 1,1
%A A335375 _Gus Wiseman_, Jun 04 2020