A337459 - OEIS

%I #11 Sep 17 2020 20:34:06

%S 7,11,13,14,19,21,25,26,28,35,37,41,42,49,50,52,56,67,69,73,74,81,82,

%T 84,97,98,100,104,112,131,133,137,138,145,146,161,162,164,168,193,194,

%U 196,200,208,224,259,261,265,266,273,274,289,290,292,321,322,324

%N Numbers k such that the k-th composition in standard order is a unimodal triple.

%C A composition of n is a finite sequence of positive integers summing to n.

%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

%C 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.

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

%F Complement of A335373 in A014311.

%e The sequence together with the corresponding triples begins:

%e       7: (1,1,1)     52: (1,2,3)    133: (5,2,1)

%e      11: (2,1,1)     56: (1,1,4)    137: (4,3,1)

%e      13: (1,2,1)     67: (5,1,1)    138: (4,2,2)

%e      14: (1,1,2)     69: (4,2,1)    145: (3,4,1)

%e      19: (3,1,1)     73: (3,3,1)    146: (3,3,2)

%e      21: (2,2,1)     74: (3,2,2)    161: (2,5,1)

%e      25: (1,3,1)     81: (2,4,1)    162: (2,4,2)

%e      26: (1,2,2)     82: (2,3,2)    164: (2,3,3)

%e      28: (1,1,3)     84: (2,2,3)    168: (2,2,4)

%e      35: (4,1,1)     97: (1,5,1)    193: (1,6,1)

%e      37: (3,2,1)     98: (1,4,2)    194: (1,5,2)

%e      41: (2,3,1)    100: (1,3,3)    196: (1,4,3)

%e      42: (2,2,2)    104: (1,2,4)    200: (1,3,4)

%e      49: (1,4,1)    112: (1,1,5)    208: (1,2,5)

%e      50: (1,3,2)    131: (6,1,1)    224: (1,1,6)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,1000],Length[stc[#]]==3&&!MatchQ[stc[#],{x_,y_,z_}/;x>y<z]&]

%Y A337460 is the non-unimodal version.

%Y A000217(n - 2) counts 3-part compositions.

%Y 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts strict 3-part compositions.

%Y A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.

%Y A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.

%Y A001523 counts unimodal compositions.

%Y A007052 counts unimodal patterns.

%Y A011782 counts unimodal permutations.

%Y A115981 counts non-unimodal compositions.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - Length is A000120.

%Y - Triples are A014311, with strict case A337453.

%Y - Sum is A070939.

%Y - Runs are counted by A124767.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%Y - Heinz number is A333219.

%Y - Combinatory separations are counted by A334030.

%Y - Non-unimodal compositions are A335373.

%Y - Non-co-unimodal compositions are A335374.

%Y Cf. A007304, A014612, A072706, A156040, A211540, A227038, A332743, A337452, A337461, A337604.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 07 2020