A225620 -id:A225620 - cp's OEIS frontend

A351009 Numbers k such that the k-th composition in standard order is a concatenation of distinct twins (x,x).

0, 3, 10, 36, 43, 58, 136, 147, 228, 528, 547, 586, 676, 904, 2080, 2115, 2186, 2347, 2362, 2696, 2707, 2788, 3600, 3658, 3748, 8256, 8323, 8458, 8740, 8747, 8762, 9352, 10768, 10787, 11144, 14368, 14474, 14984, 32896, 33027, 33290, 33828, 33835, 33850, 34963

Offset: 1

Gus Wiseman, Feb 03 2022

nonn

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 terms together with their binary expansions and standard compositions begin:
    0:           0  ()
    3:          11  (1,1)
   10:        1010  (2,2)
   36:      100100  (3,3)
   43:      101011  (2,2,1,1)
   58:      111010  (1,1,2,2)
  136:    10001000  (4,4)
  147:    10010011  (3,3,1,1)
  228:    11100100  (1,1,3,3)
  528:  1000010000  (5,5)
  547:  1000100011  (4,4,1,1)
  586:  1001001010  (3,3,2,2)
  676:  1010100100  (2,2,3,3)
  904:  1110001000  (1,1,4,4)

The case of twins (binary weight 2) is A000120.
All terms are evil numbers A001969.
The version for Heinz numbers of partitions is A062503, counted by A035457.
These compositions are counted by A032020 interspersed with 0's.
Taking singles instead of twins gives A349051.
This is the strict (distinct twins) version of A351010 and A351011.
A011782 counts compositions.
A085207 represents concatenation using standard compositions.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351014 counts distinct runs in standard compositions, see A351015.
Cf. A003242, A027383, A035363, A088218, A106356, A122134, A238279, A344604, A349054, A351005, A351007.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]], 1],0]]//Reverse;
Select[Range[0,1000], UnsameQ@@Split[stc[#]]&&And@@(#==2&)/@Length/@Split[stc[#]]&]

A353696 Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence that is already counted.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 16, 32, 43, 58, 64, 128, 256, 292, 349, 442, 512, 586, 676, 697, 826, 1024, 1210, 1338, 1393, 1394, 1396, 1594, 2048, 2186, 2234, 2618, 2696, 2785, 2786, 2792, 3130, 4096, 4282, 4410, 4666, 5178, 5569, 5570, 5572, 5576, 5584, 6202, 8192

Offset: 1

Gus Wiseman, May 22 2022

nonn

First differs from the non-consecutive version A353431 in lacking 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are a subsequence but not a consecutive subsequence.

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 terms together with their corresponding compositions begin:
    0: ()
    1: (1)
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   32: (6)
   43: (2,2,1,1)
   58: (1,1,2,2)
   64: (7)
  128: (8)
  256: (9)
  292: (3,3,3)
  349: (2,2,1,1,2,1)
  442: (1,2,1,1,2,2)
  512: (10)
  586: (3,3,2,2)
  676: (2,2,3,3)
  697: (2,2,1,1,3,1)
  826: (1,3,1,1,2,2)

Non-recursive non-consecutive for partitions: A325755, counted by A325702.
Non-consecutive: A353431, counted by A353391.
Non-consecutive for partitions: A353393, counted by A353426.
Non-recursive non-consecutive: A353402, counted by A353390.
Counted by: A353430.
Non-recursive: A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, run-lengths A333769.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, contiguous A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, multisets A225620, sets A333256.
- Runs are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Anti-runs are A333489, counted by A003242.
Cf. A032020, A114640, A181819, A228351, A329739, A318928, A325705, A329738, A333224, A353427, A353403.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]]&&yoyQ[Length/@Split[y]];
Select[Range[0,1000],yoyQ[stc[#]]&]

A350250 Numbers k such that the k-th composition in standard order is a non-alternating permutation of an initial interval of positive integers.

Original entry on oeis.org

37, 52, 549, 550, 556, 564, 581, 600, 616, 649, 657, 712, 786, 802, 836, 840, 16933, 16934, 16937, 16940, 16946, 16948, 16965, 16977, 16984, 16994, 17000, 17033, 17041, 17092, 17096, 17170, 17186, 17220, 17224, 17445, 17446, 17452, 17460, 17541, 17569, 17584

Offset: 1

Gus Wiseman, Jan 13 2022

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.

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 terms and corresponding permutations begin:
     37: (3,2,1)
     52: (1,2,3)
    549: (4,3,2,1)
    550: (4,3,1,2)
    556: (4,2,1,3)
    564: (4,1,2,3)
    581: (3,4,2,1)
    600: (3,2,1,4)
    616: (3,1,2,4)
    649: (2,4,3,1)
    657: (2,3,4,1)
    712: (2,1,3,4)
    786: (1,4,3,2)
    802: (1,3,4,2)
    836: (1,2,4,3)
    840: (1,2,3,4)
  16933: (5,4,3,2,1)

This is the non-alternating case of A333218.
This is the restriction of A345168 to permutations, complement A345167.
These partitions are counted by A348615, complement A001250.
A003242 counts anti-run compositions, patterns A005649.
A025047 counts alternating compositions, directed A025048/A025049.
A345192 counts non-alternating compositions.
A345194 counts alternating patterns, complement A350252.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994, strict A333256.
- Weakly increasing compositions (multisets) are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Anti-run compositions are A333489, complement A348612.
Cf. A008965, A059893, A164894, A246534, A333217, A344605, A345162, A350251, A345163, A345171, A345172, A348613.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[0,1000],(Sort[stc[#]]==Range[Length[stc[#]]]&&!wigQ[stc[#]])&]

A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.

Original entry on oeis.org

13, 22, 25, 27, 29, 38, 41, 44, 45, 46, 49, 50, 51, 53, 54, 55, 57, 59, 61, 70, 76, 77, 78, 81, 82, 83, 86, 88, 89, 90, 91, 92, 93, 94, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 134, 140, 141, 142

Offset: 1

Gus Wiseman, Sep 18 2024

nonn

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 terms and corresponding compositions begin:
  13: (1,2,1)
  22: (2,1,2)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  38: (3,1,2)
  41: (2,3,1)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)

Eric Weisstein's World of Mathematics, Unimodal Sequence.

The version for run-lengths of compositions is A332833.
Compositions of this type are counted by A332834, complement maybe A329398.
A001523 counts unimodal compositions, ranks too dense.
A011782 counts compositions.
A114994 ranks weakly decreasing compositions, complement A335485.
A115981 counts non-unimodal compositions, ranked by A335373.
A225620 ranks weakly increasing compositions, complement A335486.
A238130, A238279, A333755 count compositions by number of runs.
A332835 counts compositions with weakly incr. or weakly decr. run-lengths.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564.
- Ranks of constant compositions are A272919.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
Cf. A001511, A002051, A065120, A329744, A332745, A332836, A332870, A333218.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!LessEqual@@stc[#]&&!GreaterEqual@@stc[#]&]

Intersection of A335485 and A335486.

A349152 Standard composition numbers of compositions into divisors. Numbers k such that all parts of the k-th composition in standard order are divisors of the sum of parts.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 11, 13, 14, 15, 16, 31, 32, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 127, 128, 136, 138, 139, 141, 142, 143, 162, 163, 168, 170, 171, 173, 174, 175, 177, 181, 182, 183, 184

Offset: 1

Gus Wiseman, Nov 15 2021

nonn

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 terms and corresponding compositions begin:
      0: ()              36: (3,3)           54: (1,2,1,2)
      1: (1)             37: (3,2,1)         55: (1,2,1,1,1)
      2: (2)             38: (3,1,2)         57: (1,1,3,1)
      3: (1,1)           39: (3,1,1,1)       58: (1,1,2,2)
      4: (3)             41: (2,3,1)         59: (1,1,2,1,1)
      7: (1,1,1)         42: (2,2,2)         60: (1,1,1,3)
      8: (4)             43: (2,2,1,1)       61: (1,1,1,2,1)
     10: (2,2)           44: (2,1,3)         62: (1,1,1,1,2)
     11: (2,1,1)         45: (2,1,2,1)       63: (1,1,1,1,1,1)
     13: (1,2,1)         46: (2,1,1,2)       64: (7)
     14: (1,1,2)         47: (2,1,1,1,1)    127: (1,1,1,1,1,1,1)
     15: (1,1,1,1)       50: (1,3,2)        128: (8)
     16: (5)             51: (1,3,1,1)      136: (4,4)
     31: (1,1,1,1,1)     52: (1,2,3)        138: (4,2,2)
     32: (6)             53: (1,2,2,1)      139: (4,2,1,1)

Looking at length instead of parts gives A096199.
These composition are counted by A100346.
A version counting subsets instead of compositions is A125297.
An unordered version is A326841, counted by A018818.
A011782 counts compositions.
A316413 ranks partitions with sum divisible by length, counted by A067538.
A319333 ranks partitions with sum equal to lcm, counted by A074761.
Cf. A003242, A056239, A238279, A326836, A326842.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Permutations are ranked by A333218.
- Relatively prime compositions are ranked by A291166*, complement A291165.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],#==0||Divisible[Total[stc[#]],LCM@@stc[#]]&]

A373120 Number of distinct possible binary ranks of integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 33, 43, 55, 70, 89, 109, 136, 167, 206, 251, 306, 371, 445, 535, 639, 759, 904, 1069, 1262, 1489, 1747, 2047, 2390, 2784, 3237, 3754, 4350, 5027, 5798, 6680, 7671, 8808, 10091, 11543, 13190, 15040, 17128, 19477, 22118

Offset: 0

Gus Wiseman, May 26 2024

nonn

			The partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1), with respective binary ranks 8, 5, 4, 4, 4, so a(4) = 3.

The strict case is A000009.
A048675 gives binary rank of prime indices, distinct A087207.
A118462 lists binary ranks of strict integer partitions, row sums A372888.
A277905 groups all positive integers by binary rank of prime indices.
A372890 adds up binary ranks of integer partitions.
Binary indices (A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- max A029837 or A070939, opposite A070940
- sum A029931, product A096111
- reverse A272020
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000041, A018819, A158704, A158705, A225620, A231204, A372688.

Table[Length[Union[Total[2^(#-1)]&/@IntegerPartitions[n]]],{n,0,15}]

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A351009 Numbers k such that the k-th composition in standard order is a concatenation of distinct twins (x,x).

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A353696 Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence that is already counted.

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A350250 Numbers k such that the k-th composition in standard order is a non-alternating permutation of an initial interval of positive integers.

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A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.

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A349152 Standard composition numbers of compositions into divisors. Numbers k such that all parts of the k-th composition in standard order are divisors of the sum of parts.

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A373120 Number of distinct possible binary ranks of integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

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