A335374 -id:A335374 - cp's OEIS frontend

A335373 Numbers k such that the k-th composition in standard order (A066099) is not unimodal.

22, 38, 44, 45, 46, 54, 70, 76, 77, 78, 86, 88, 89, 90, 91, 92, 93, 94, 102, 108, 109, 110, 118, 134, 140, 141, 142, 148, 150, 152, 153, 154, 155, 156, 157, 158, 166, 172, 173, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 198

Offset: 1

Gus Wiseman, Jun 03 2020

nonn

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

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.

			The sequence together with the corresponding compositions begins:
  22: (2,1,2)
  38: (3,1,2)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  54: (1,2,1,2)
  70: (4,1,2)
  76: (3,1,3)
  77: (3,1,2,1)
  78: (3,1,1,2)
  86: (2,2,1,2)
  88: (2,1,4)
  89: (2,1,3,1)
  90: (2,1,2,2)
  91: (2,1,2,1,1)
  92: (2,1,1,3)
  93: (2,1,1,2,1)
  94: (2,1,1,1,2)

The dual version (non-co-unimodal compositions) is A335374.
The case that is not co-unimodal either is A335375.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Partitions with non-unimodal 0-appended first differences are A332284.
Non-unimodal permutations of the multiset of prime indices of n are A332671.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A072704, A332281, A332286, A332287, A332639, A332642, A332669, A332672.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,200],!unimodQ[stc[#]]&]

A337459 Numbers k such that the k-th composition in standard order is a unimodal triple.

Original entry on oeis.org

7, 11, 13, 14, 19, 21, 25, 26, 28, 35, 37, 41, 42, 49, 50, 52, 56, 67, 69, 73, 74, 81, 82, 84, 97, 98, 100, 104, 112, 131, 133, 137, 138, 145, 146, 161, 162, 164, 168, 193, 194, 196, 200, 208, 224, 259, 261, 265, 266, 273, 274, 289, 290, 292, 321, 322, 324

Offset: 1

Gus Wiseman, Sep 07 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
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.

			The sequence together with the corresponding triples begins:
      7: (1,1,1)     52: (1,2,3)    133: (5,2,1)
     11: (2,1,1)     56: (1,1,4)    137: (4,3,1)
     13: (1,2,1)     67: (5,1,1)    138: (4,2,2)
     14: (1,1,2)     69: (4,2,1)    145: (3,4,1)
     19: (3,1,1)     73: (3,3,1)    146: (3,3,2)
     21: (2,2,1)     74: (3,2,2)    161: (2,5,1)
     25: (1,3,1)     81: (2,4,1)    162: (2,4,2)
     26: (1,2,2)     82: (2,3,2)    164: (2,3,3)
     28: (1,1,3)     84: (2,2,3)    168: (2,2,4)
     35: (4,1,1)     97: (1,5,1)    193: (1,6,1)
     37: (3,2,1)     98: (1,4,2)    194: (1,5,2)
     41: (2,3,1)    100: (1,3,3)    196: (1,4,3)
     42: (2,2,2)    104: (1,2,4)    200: (1,3,4)
     49: (1,4,1)    112: (1,1,5)    208: (1,2,5)
     50: (1,3,2)    131: (6,1,1)    224: (1,1,6)

Eric Weisstein's World of Mathematics, Unimodal Sequence

A337460 is the non-unimodal version.
A000217(n - 2) counts 3-part compositions.
6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts strict 3-part compositions.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.
A001523 counts unimodal compositions.
A007052 counts unimodal patterns.
A011782 counts unimodal permutations.
A115981 counts non-unimodal compositions.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Triples are A014311, with strict case A337453.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Heinz number is A333219.
- Combinatory separations are counted by A334030.
- Non-unimodal compositions are A335373.
- Non-co-unimodal compositions are A335374.
Cf. A007304, A014612, A072706, A156040, A211540, A227038, A332743, A337452, A337461, A337604.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Length[stc[#]]==3&&!MatchQ[stc[#],{x_,y_,z_}/;x>y

Complement of A335373 in A014311.

A337460 Numbers k such that the k-th composition in standard order is a non-unimodal triple.

Original entry on oeis.org

22, 38, 44, 70, 76, 88, 134, 140, 148, 152, 176, 262, 268, 276, 280, 296, 304, 352, 518, 524, 532, 536, 552, 560, 592, 608, 704, 1030, 1036, 1044, 1048, 1064, 1072, 1096, 1104, 1120, 1184, 1216, 1408, 2054, 2060, 2068, 2072, 2088, 2096, 2120, 2128, 2144, 2192

Offset: 1

Gus Wiseman, Sep 18 2020

nonn

These are triples matching the pattern (2,1,2), (3,1,2), or (2,1,3).
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
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.

			The sequence together with the corresponding triples begins:
      22: (2,1,2)     296: (3,2,4)    1048: (6,1,4)
      38: (3,1,2)     304: (3,1,5)    1064: (5,2,4)
      44: (2,1,3)     352: (2,1,6)    1072: (5,1,5)
      70: (4,1,2)     518: (7,1,2)    1096: (4,3,4)
      76: (3,1,3)     524: (6,1,3)    1104: (4,2,5)
      88: (2,1,4)     532: (5,2,3)    1120: (4,1,6)
     134: (5,1,2)     536: (5,1,4)    1184: (3,2,6)
     140: (4,1,3)     552: (4,2,4)    1216: (3,1,7)
     148: (3,2,3)     560: (4,1,5)    1408: (2,1,8)
     152: (3,1,4)     592: (3,2,5)    2054: (9,1,2)
     176: (2,1,5)     608: (3,1,6)    2060: (8,1,3)
     262: (6,1,2)     704: (2,1,7)    2068: (7,2,3)
     268: (5,1,3)    1030: (8,1,2)    2072: (7,1,4)
     276: (4,2,3)    1036: (7,1,3)    2088: (6,2,4)
     280: (4,1,4)    1044: (6,2,3)    2096: (6,1,5)

Eric Weisstein's World of Mathematics, Unimodal Sequence
Gus Wiseman, Statistics, classes, and transformations of standard compositions

A000212 counts unimodal triples.
A000217(n - 2) counts 3-part compositions.
A001399(n - 3) counts 3-part partitions.
A001399(n - 6) counts 3-part strict partitions.
A001399(n - 6)*2 counts non-unimodal 3-part strict compositions.
A001399(n - 6)*4 counts unimodal 3-part strict compositions.
A001399(n - 6)*6 counts 3-part strict compositions.
A001523 counts unimodal compositions.
A001840 counts non-unimodal triples.
A059204 counts non-unimodal permutations.
A115981 counts non-unimodal compositions.
A328509 counts non-unimodal patterns.
A337459 ranks unimodal triples.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Triples are A014311.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Heinz number is A333219.
- Non-unimodal compositions are A335373.
- Non-co-unimodal compositions are A335374.
- Strict triples are A337453.
Cf. A007304, A014612, A069905, A072706, A156040, A211540, A227038, A332743, A337461, A337604.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Length[stc[#]]==3&&MatchQ[stc[#],{x_,y_,z_}/;x>y

Intersection of A014311 and A335373.

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

Original entry on oeis.org

45, 54, 77, 89, 91, 93, 102, 108, 109, 110, 118, 141, 153, 155, 157, 166, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189, 198, 204, 205, 206, 214, 216, 217, 218, 219, 220, 221, 222, 230, 236, 237, 238, 246, 269, 281, 283, 285, 297, 301, 305, 306, 307, 309

Offset: 1

Gus Wiseman, Jun 04 2020

nonn

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.

			The sequence together with the corresponding compositions begins:
   45: (2,1,2,1)
   54: (1,2,1,2)
   77: (3,1,2,1)
   89: (2,1,3,1)
   91: (2,1,2,1,1)
   93: (2,1,1,2,1)
  102: (1,3,1,2)
  108: (1,2,1,3)
  109: (1,2,1,2,1)
  110: (1,2,1,1,2)
  118: (1,1,2,1,2)
  141: (4,1,2,1)
  153: (3,1,3,1)
  155: (3,1,2,1,1)
  157: (3,1,1,2,1)
  166: (2,3,1,2)
  173: (2,2,1,2,1)
  177: (2,1,4,1)
  178: (2,1,3,2)
  179: (2,1,3,1,1)

Non-unimodal compositions are ranked by A335373.
Non-co-unimodal compositions are ranked by A335374.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Co-unimodal compositions are A332578.
Numbers with non-co-unimodal unsorted prime signature are A332642.
Non-co-unimodal compositions are A332669.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!unimodQ[stc[#]]&&!unimodQ[-stc[#]]&]

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A335373 Numbers k such that the k-th composition in standard order (A066099) is not unimodal.

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A337459 Numbers k such that the k-th composition in standard order is a unimodal triple.

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A337460 Numbers k such that the k-th composition in standard order is a non-unimodal triple.

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

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