A337460 - OEIS

%I #6 Sep 21 2020 08:38:35

%S 22,38,44,70,76,88,134,140,148,152,176,262,268,276,280,296,304,352,

%T 518,524,532,536,552,560,592,608,704,1030,1036,1044,1048,1064,1072,

%U 1096,1104,1120,1184,1216,1408,2054,2060,2068,2072,2088,2096,2120,2128,2144,2192

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

%C These are triples matching the pattern (2,1,2), (3,1,2), or (2,1,3).

%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>

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%F Intersection of A014311 and A335373.

%e The sequence together with the corresponding triples begins:

%e       22: (2,1,2)     296: (3,2,4)    1048: (6,1,4)

%e       38: (3,1,2)     304: (3,1,5)    1064: (5,2,4)

%e       44: (2,1,3)     352: (2,1,6)    1072: (5,1,5)

%e       70: (4,1,2)     518: (7,1,2)    1096: (4,3,4)

%e       76: (3,1,3)     524: (6,1,3)    1104: (4,2,5)

%e       88: (2,1,4)     532: (5,2,3)    1120: (4,1,6)

%e      134: (5,1,2)     536: (5,1,4)    1184: (3,2,6)

%e      140: (4,1,3)     552: (4,2,4)    1216: (3,1,7)

%e      148: (3,2,3)     560: (4,1,5)    1408: (2,1,8)

%e      152: (3,1,4)     592: (3,2,5)    2054: (9,1,2)

%e      176: (2,1,5)     608: (3,1,6)    2060: (8,1,3)

%e      262: (6,1,2)     704: (2,1,7)    2068: (7,2,3)

%e      268: (5,1,3)    1030: (8,1,2)    2072: (7,1,4)

%e      276: (4,2,3)    1036: (7,1,3)    2088: (6,2,4)

%e      280: (4,1,4)    1044: (6,2,3)    2096: (6,1,5)

%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 A000212 counts unimodal triples.

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

%Y A001399(n - 3) counts 3-part partitions.

%Y A001399(n - 6) counts 3-part strict partitions.

%Y A001399(n - 6)*2 counts non-unimodal 3-part strict compositions.

%Y A001399(n - 6)*4 counts unimodal 3-part strict compositions.

%Y A001399(n - 6)*6 counts 3-part strict compositions.

%Y A001523 counts unimodal compositions.

%Y A001840 counts non-unimodal triples.

%Y A059204 counts non-unimodal permutations.

%Y A115981 counts non-unimodal compositions.

%Y A328509 counts non-unimodal patterns.

%Y A337459 ranks unimodal triples.

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

%Y - Length is A000120.

%Y - Triples are A014311.

%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 - Non-unimodal compositions are A335373.

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

%Y - Strict triples are A337453.

%Y Cf. A007304, A014612, A069905, A072706, A156040, A211540, A227038, A332743, A337461, A337604.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 18 2020