A325676 -id:A325676 - cp's OEIS frontend

A335458 Number of normal patterns contiguously matched by the n-th composition in standard order (A066099).

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 5, 5, 5, 2, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 7, 8, 9, 3, 5, 5, 8, 4, 8, 7, 11, 5, 8, 7, 11, 7, 11, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 5, 7, 8, 9, 3, 5, 5, 8, 5, 7

Offset: 0

Gus Wiseman, Jun 21 2020

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.

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(180) = 7 patterns are: (), (1), (1,2), (2,1), (1,2,3), (2,1,2), (2,1,2,3).

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The non-contiguous version is A335454.
Summing over indices with binary length n gives A335457(n).
The nonempty version is A335474.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Cf. A108917, A124767, A124770, A181796, A269134, A333218, A333222, A333628, A334030, A335456.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@ReplaceList[stc[n],{_,s___,_}:>{s}]]],{n,0,30}]

a(n) = A335474(n) + 1.

A334299 Number of distinct subsequences (not necessarily contiguous) of compositions in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 6, 4, 7, 6, 5, 2, 4, 4, 6, 4, 6, 7, 8, 4, 7, 6, 10, 6, 10, 8, 6, 2, 4, 4, 6, 3, 8, 8, 8, 4, 8, 4, 9, 8, 12, 11, 10, 4, 7, 8, 10, 8, 11, 12, 13, 6, 10, 9, 14, 8, 13, 10, 7, 2, 4, 4, 6, 4, 8, 8, 8, 4, 6, 6, 12, 7, 14, 12, 10, 4

Offset: 0

Gus Wiseman, Jun 01 2020

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.

			Triangle begins:
  1
  2
  2 3
  2 4 4 4
  2 4 3 6 4 7 6 5
  2 4 4 6 4 6 7 8 4 7 6 10 6 10 8 6
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The n-th column below lists the STC-numbers of the subsequences of the composition with STC-number n:
  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
     0  0  1  0  2  2  3  0  4   2   5   4   6   6   7
           0     1  1  1     1   0   3   1   5   3   3
                 0  0  0     0       2   0   3   2   1
                                     1       2   1   0
                                     0       1   0
                                             0

Row lengths are A011782.
Looking only at contiguous subsequences gives A124771.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Contiguous positive subsequence-sums are counted by A333224.
Contiguous subsequence-sums are counted by A333257.
Disallowing empty subsequences gives A334300.
Subsequence-sums are counted by A334968.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325676, A334967.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Subsets[stc[n]]]],{n,0,100}]

a(n) = A334300(n) + 1.

A333223 Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 31, 32, 33, 34, 35, 36, 40, 41, 42, 48, 50, 56, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 84, 85, 88, 96, 98, 100, 104, 106, 112, 120, 127, 128, 129, 130, 131, 132, 133

Offset: 1

Gus Wiseman, Mar 17 2020

nonn

The k-th composition in standard order (row k of 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.

			The list of terms together with the corresponding compositions begins:
    0: ()            21: (2,2,1)           65: (6,1)
    1: (1)           24: (1,4)             66: (5,2)
    2: (2)           26: (1,2,2)           67: (5,1,1)
    3: (1,1)         28: (1,1,3)           68: (4,3)
    4: (3)           31: (1,1,1,1,1)       69: (4,2,1)
    5: (2,1)         32: (6)               70: (4,1,2)
    6: (1,2)         33: (5,1)             71: (4,1,1,1)
    7: (1,1,1)       34: (4,2)             72: (3,4)
    8: (4)           35: (4,1,1)           73: (3,3,1)
    9: (3,1)         36: (3,3)             74: (3,2,2)
   10: (2,2)         40: (2,4)             80: (2,5)
   12: (1,3)         41: (2,3,1)           81: (2,4,1)
   15: (1,1,1,1)     42: (2,2,2)           84: (2,2,3)
   16: (5)           48: (1,5)             85: (2,2,2,1)
   17: (4,1)         50: (1,3,2)           88: (2,1,4)
   18: (3,2)         56: (1,1,4)           96: (1,6)
   19: (3,1,1)       63: (1,1,1,1,1,1)     98: (1,4,2)
   20: (2,3)         64: (7)              100: (1,3,3)

Distinct subsequences are counted by A124770 and A124771.
A superset of A333222, counted by A169942, with partition case A325768.
These compositions are counted by A325676.
A version for partitions is A325769, with Heinz numbers A325778.
The number of distinct positive subsequence-sums is A333224.
The number of distinct subsequence-sums is A333257.
Numbers whose binary indices are a strict knapsack partition are A059519.
Knapsack partitions are counted by A108917, with strict case A275972.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Maximal Golomb rulers are counted by A325683.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295 A325779, A233564, A325680, A325687, A325770, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Total/@Union[ReplaceList[stc[#],{_,s__,_}:>{s}]]&]

A325683 Number of maximal Golomb rulers of length n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 2, 6, 8, 18, 16, 24, 20, 28, 42, 76, 100, 138, 168, 204, 194, 272, 276, 450, 588, 808, 992, 1578, 1612, 1998, 2166, 2680, 2732, 3834, 3910, 5716, 6818, 9450, 10524, 15504, 16640, 22268, 23754, 30430, 31498, 40644, 40294, 52442, 56344, 72972, 77184

Offset: 0

Gus Wiseman, May 13 2019

nonn

A Golomb ruler of length n is a subset of {0..n} containing 0 and n and such that every pair of distinct terms has a different difference up to sign.

Also the number of minimal (most refined) compositions of n such that every restriction to a subinterval has a different sum.

			The a(1) = 1 through a(8) = 8 maximal Golomb rulers:
  {0,1}  {0,2}  {0,1,3}  {0,1,4}  {0,1,5}  {0,1,4,6}  {0,1,3,7}  {0,1,3,8}
                {0,2,3}  {0,3,4}  {0,2,5}  {0,2,5,6}  {0,1,5,7}  {0,1,5,8}
                                  {0,3,5}             {0,2,3,7}  {0,1,6,8}
                                  {0,4,5}             {0,2,6,7}  {0,2,3,8}
                                                      {0,4,5,7}  {0,2,7,8}
                                                      {0,4,6,7}  {0,3,7,8}
                                                                 {0,5,6,8}
                                                                 {0,5,7,8}
The a(1) = 1 through a(10) = 16 minimal compositions:
  (1)  (2)  (12)  (13)  (14)  (132)  (124)  (125)  (126)  (127)
            (21)  (31)  (23)  (231)  (142)  (143)  (135)  (136)
                        (32)         (214)  (152)  (153)  (154)
                        (41)         (241)  (215)  (162)  (163)
                                     (412)  (251)  (216)  (172)
                                     (421)  (341)  (234)  (217)
                                            (512)  (243)  (253)
                                            (521)  (261)  (271)
                                                   (315)  (316)
                                                   (324)  (352)
                                                   (342)  (361)
                                                   (351)  (451)
                                                   (423)  (613)
                                                   (432)  (631)
                                                   (513)  (712)
                                                   (531)  (721)
                                                   (612)
                                                   (621)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Golomb Ruler.

Rightmost column of A325677.
Cf. A000079, A103295, A108917, A143823, A169942, A276024.
Cf. A325676, A325678, A325679, A325681, A325687.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Accumulate/@Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&]]],{n,0,15}]

a(21)-a(50) from Fausto A. C. Cariboni, Feb 22 2022

A333222 Numbers k such that every restriction of the k-th composition in standard order to a subinterval has a different sum.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 16, 17, 18, 20, 24, 32, 33, 34, 40, 41, 48, 50, 64, 65, 66, 68, 69, 70, 72, 80, 81, 88, 96, 98, 104, 128, 129, 130, 132, 133, 134, 144, 145, 160, 161, 176, 192, 194, 196, 208, 256, 257, 258, 260, 261, 262, 264, 265, 268, 272, 274

Offset: 1

Gus Wiseman, Mar 17 2020

nonn

Also numbers whose binary indices together with 0 define a Golomb ruler.

The k-th composition in standard order (row k of 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.

			The list of terms together with the corresponding compositions begins:
    0: ()        41: (2,3,1)    130: (6,2)      262: (6,1,2)
    1: (1)       48: (1,5)      132: (5,3)      264: (5,4)
    2: (2)       50: (1,3,2)    133: (5,2,1)    265: (5,3,1)
    4: (3)       64: (7)        134: (5,1,2)    268: (5,1,3)
    5: (2,1)     65: (6,1)      144: (3,5)      272: (4,5)
    6: (1,2)     66: (5,2)      145: (3,4,1)    274: (4,3,2)
    8: (4)       68: (4,3)      160: (2,6)      276: (4,2,3)
    9: (3,1)     69: (4,2,1)    161: (2,5,1)    288: (3,6)
   12: (1,3)     70: (4,1,2)    176: (2,1,5)    289: (3,5,1)
   16: (5)       72: (3,4)      192: (1,7)      290: (3,4,2)
   17: (4,1)     80: (2,5)      194: (1,5,2)    296: (3,2,4)
   18: (3,2)     81: (2,4,1)    196: (1,4,3)    304: (3,1,5)
   20: (2,3)     88: (2,1,4)    208: (1,2,5)    320: (2,7)
   24: (1,4)     96: (1,6)      256: (9)        321: (2,6,1)
   32: (6)       98: (1,4,2)    257: (8,1)      324: (2,4,3)
   33: (5,1)    104: (1,2,4)    258: (7,2)      328: (2,3,4)
   34: (4,2)    128: (8)        260: (6,3)      352: (2,1,6)
   40: (2,4)    129: (7,1)      261: (6,2,1)    384: (1,8)

Wikipedia, Golomb ruler

A subset of A233564.
Also a subset of A333223.
These compositions are counted by A169942 and A325677.
The number of distinct nonzero subsequence-sums is A333224.
The number of distinct subsequence-sums is A333257.
Lengths of optimal Golomb rulers are A003022.
Inequivalent optimal Golomb rulers are counted by A036501.
Complete rulers are A103295, with perfect case A103300.
Knapsack partitions are counted by A108917, with strict case A275972.
Distinct subsequences are counted by A124770 and A124771.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Knapsack compositions are counted by A325676.
Maximal Golomb rulers are counted by A325683.
Cf. A000120, A029931, A048793, A066099, A070939, A228351, A295235, A325678, A325680, A325768, A325779, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,300],UnsameQ@@ReplaceList[stc[#],{_,s__,_}:>Plus[s]]&]

A325677 Irregular triangle read by rows where T(n,k) is the number of Golomb rulers of length n with k + 1 marks, k > 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 1, 4, 2, 1, 6, 6, 1, 6, 8, 1, 8, 18, 1, 8, 16, 1, 10, 30, 4, 1, 10, 34, 14, 1, 12, 48, 28, 1, 12, 48, 42, 1, 14, 72, 76, 1, 14, 72, 100, 1, 16, 96, 160, 8, 1, 16, 98, 190, 8, 1, 18, 126, 284, 40, 1, 18, 128, 316, 70

Offset: 1

Gus Wiseman, May 13 2019

Also the number of length-k compositions of n such that every restriction to a subinterval has a different sum. A composition of n is a finite sequence of positive integers summing to n.

			Triangle begins:
   1
   1
   1   2
   1   2
   1   4
   1   4   2
   1   6   6
   1   6   8
   1   8  18
   1   8  16
   1  10  30   4
   1  10  34  14
   1  12  48  28
   1  12  48  42
   1  14  72  76
   1  14  72 100
   1  16  96 160   8
   1  16  98 190   8
   1  18 126 284  40
   1  18 128 316  70
Row n = 8 counts the following rulers:
  {0,8}  {0,1,8}  {0,1,3,8}
         {0,2,8}  {0,1,5,8}
         {0,3,8}  {0,1,6,8}
         {0,5,8}  {0,2,3,8}
         {0,6,8}  {0,2,7,8}
         {0,7,8}  {0,3,7,8}
                  {0,5,6,8}
                  {0,5,7,8}
and the following compositions:
  (8)  (17)  (125)
       (26)  (143)
       (35)  (152)
       (53)  (215)
       (62)  (251)
       (71)  (341)
             (512)
             (521)

Eric Weisstein's World of Mathematics, Golomb Ruler.

Row sums are A169942.
Row lengths are A325678(n) = A143824(n + 1) - 1.
Column k = 2 is A052928.
Column k = 3 is A325686.
Rightmost column is A325683.
Cf. A000079, A007318, A103295, A108917, A143823, A325676, A325679, A325687.

DeleteCases[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&]],{n,15},{k,n}],0,{2}]

A143824 Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12

Offset: 0

John W. Layman, Sep 02 2008

When the set {x(1),x(2),...,x(k)} satisfies the property that all differences |x(i)-x(j)| are distinct (or alternately, all the sums are distinct), then it is called a Sidon set. So a(n) is the maximum cardinality of a dense Sidon subset of {1,2,...,n}. - Sayan Dutta, Aug 29 2024
See A143823 for the number of subsets of {1, 2, ..., n} with the required property.
See A003022 (and A227590) for the values of n such that a(n+1) > a(n). - Boris Bukh, Jul 28 2013
Can be formulated as an integer linear program: maximize sum {i = 1 to n} z[i] subject to z[i] + z[j] - 1 <= y[i,j] for all i < j, sum {i = 1 to n - d} y[i,i+d] <= 1 for d = 1 to n - 1, z[i] in {0,1} for all i, y[i,j] in {0,1} for all i < j. - Rob Pratt, Feb 09 2010
If the zeroth term is removed, the run-lengths are A270813 with 1 prepended. - Gus Wiseman, Jun 07 2019

			For n = 4, {1, 2, 4} is a subset of {1, 2, 3, 4} with distinct differences 2 - 1 = 1, 4 - 1 = 3, 4 - 2 = 2 between pairs of elements and no larger set has the required property; so a(4) = 3.
From _Gus Wiseman_, Jun 07 2019: (Start)
Examples of subsets realizing each largest size are:
   0: {}
   1: {1}
   2: {1,2}
   3: {2,3}
   4: {1,3,4}
   5: {2,4,5}
   6: {3,5,6}
   7: {1,3,6,7}
   8: {2,4,7,8}
   9: {3,5,8,9}
  10: {4,6,9,10}
  11: {5,7,10,11}
  12: {1,4,5,10,12}
  13: {2,5,6,11,13}
  14: {3,6,7,12,14}
  15: {4,7,8,13,15}
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..500
Thomas Bloom, Problem 30, Problem 43, Problem 155, and Problem 861, Erdős Problems.
Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.
Terence Tao, Erdős problem database, see nos. 30, 43, 155, 861.
Wikipedia, Sidon sequence.
Wikipedia, Golomb ruler
Balogh, J., Füredi, Z., & Roy, S. (2023), An Upper Bound on the Size of Sidon Sets, The American Mathematical Monthly, 130(5), 437-445.

Cf. A143823, A003002, A003022, A227590.

Cf. A108917, A169942, A270813, A325676, A325677, A325678, A325683, A325687.

Table[Length[Last[Select[Subsets[Range[n]],UnsameQ@@Subtract@@@Subsets[#,{2}]&]]],{n,0,15}] (* Gus Wiseman, Jun 07 2019 *)

For n > 1, a(n) = A325678(n - 1) + 1. - Gus Wiseman, Jun 07 2019
From Sayan Dutta, Aug 29 2024: (Start)
a(n) < n^(1/2) + 0.998*n^(1/4) for sufficiently large n (see Balogh et. al. link).
It is conjectured by Erdos (for $500) that a(n) < n^(1/2) + o(n^e) for all e>0. (End)

More terms from Rob Pratt, Feb 09 2010
a(41)-a(60) from Alois P. Heinz, Sep 14 2011
More terms and b-file from N. J. A. Sloane, Apr 08 2016 using data from A003022.

A325768 Number of integer partitions of n for which every restriction to a subinterval has a different sum.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 12, 15, 15, 23, 22, 29, 32, 40, 42, 55, 56, 71, 75, 92, 100, 124, 128, 152, 167, 198, 212, 255, 269, 315, 343, 392, 428, 501, 529, 615, 665, 757, 812, 937, 1002, 1142, 1238, 1385, 1490, 1701, 1808, 2038, 2200, 2476

Offset: 0

Gus Wiseman, May 21 2019

nonn

Also the number of Golomb rulers of length n whose consecutive marks are separated by weakly decreasing distances.

The Heinz numbers of these partitions are given by A325779.

			The a(1) = 1 through a(9) = 8 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)
                        (41)  (51)  (52)   (62)   (63)
                                    (61)   (71)   (72)
                                    (421)  (521)  (81)
                                                  (432)
                                                  (531)
                                                  (621)

Cf. A000041, A002033, A103300, A143823, A169942, A325676, A325687, A325769, A325779.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&]],{n,0,30}]

A325687 Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 0, 1, 1, 4, 4, 0, 1, 1, 5, 5, 0, 0, 1, 1, 6, 12, 4, 0, 0, 1, 1, 7, 12, 5, 0, 0, 0, 1, 1, 8, 25, 8, 4, 0, 0, 0, 1, 1, 9, 24, 12, 3, 0, 0, 0, 0, 1, 1, 10, 40, 32, 8, 4, 0, 0, 0, 0, 1, 1, 11, 41, 41, 6, 3, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, May 13 2019

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

			The distinct consecutive subsequences of (1,1,3,3) are (1), (1,1), (3), (1,3), (1,1,3), (3,3), (1,3,3), (1,1,3,3), all of which have different sums, so (1,1,3,3) is counted under a(8).
Triangle begins:
  1
  1  1
  1  2  1
  1  3  0  1
  1  4  4  0  1
  1  5  5  0  0  1
  1  6 12  4  0  0  1
  1  7 12  5  0  0  0  1
  1  8 25  8  4  0  0  0  1
  1  9 24 12  3  0  0  0  0  1
  1 10 40 32  8  4  0  0  0  0  1
  1 11 41 41  6  3  0  0  0  0  0  1
  1 12 60 76 14  4  4  0  0  0  0  0  1
  1 13 60 88 16  6  3  0  0  0  0  0  0  1
Row n = 8 counts the following compositions:
  (8)  (17)  (116)  (1115)  (11111111)
       (26)  (125)  (1133)
       (35)  (143)  (2222)
       (44)  (152)  (3311)
       (53)  (215)  (5111)
       (62)  (233)
       (71)  (251)
             (332)
             (341)
             (512)
             (521)
             (611)

Row sums are A325676.
Column k = 2 is A000027.
Column k = 3 is A325688.
Cf. A000079, A007318, A048004, A108917, A143823, A169942, A266223, A325592, A325680, A325685.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,15},{k,n}]

A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 5, 4, 5, 5, 5, 2, 4, 4, 6, 4, 6, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 2, 4, 4, 6, 3, 7, 7, 7, 4, 7, 4, 7, 7, 7, 7, 7, 4, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 2, 4, 4, 6, 4, 8, 8, 8, 4, 6, 6, 8, 6, 8, 8, 8, 4, 8, 6, 8, 6, 8, 8

Offset: 0

Gus Wiseman, Jun 02 2020

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 139th composition is (4,2,1,1), with possible sums of subsequences {0,1,2,3,4,5,6,7,8}, so a(139) = 9.
Triangle begins:
  1
  2
  2 3
  2 4 4 4
  2 4 3 5 4 5 5 5
  2 4 4 6 4 6 6 6 4 6 6 6 6 6 6 6
  2 4 4 6 3 7 7 7 4 7 4 7 7 7 7 7 4 6 7 7 7 7 7 7 6 7 7 7 7 7 7 7

Row lengths are A011782.
Dominated by A124771 (number of contiguous subsequences).
Dominates A333257 (the contiguous case).
Dominated by A334299 (number of subsequences).
Golomb rulers are counted by A169942 and ranked by A333222.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and ranked by A299702
Knapsack compositions are counted by A325676 and ranked by A333223.
Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
Knapsack compositions are counted by A334268 and ranked by A334967.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325769, A325770, A325778, A334300, A335279.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Total/@Subsets[stc[n]]]],{n,0,100}]

a(n) = A299701(A333219(n)).

cp's OEIS Frontend

A335458 Number of normal patterns contiguously matched by the n-th composition in standard order (A066099).

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A334299 Number of distinct subsequences (not necessarily contiguous) of compositions in standard order (A066099).

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A333223 Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.

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A325683 Number of maximal Golomb rulers of length n.

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A333222 Numbers k such that every restriction of the k-th composition in standard order to a subinterval has a different sum.

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A325677 Irregular triangle read by rows where T(n,k) is the number of Golomb rulers of length n with k + 1 marks, k > 0.

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A143824 Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.

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A325768 Number of integer partitions of n for which every restriction to a subinterval has a different sum.

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