A103295 -id:A103295 - cp's OEIS frontend

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

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]]&]

A103294 Triangle T, read by rows: T(n,k) = number of complete rulers with length n and k segments (n >= 0, k >= 0).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 3, 1, 0, 0, 0, 4, 4, 1, 0, 0, 0, 2, 9, 5, 1, 0, 0, 0, 0, 12, 14, 6, 1, 0, 0, 0, 0, 8, 27, 20, 7, 1, 0, 0, 0, 0, 4, 40, 48, 27, 8, 1, 0, 0, 0, 0, 0, 38, 90, 75, 35, 9, 1, 0, 0, 0, 0, 0, 30, 134, 166, 110, 44, 10, 1, 0, 0, 0, 0, 0, 14, 166, 311, 277, 154, 54, 11, 1

Offset: 0

Peter Luschny, Feb 28 2005

If n=k then T(n,k)=1.
A sparse ruler, or simply a ruler, is a strict increasing finite sequence of nonnegative integers starting from 0 called marks.
A segment of a ruler is the space between two adjacent marks. The number of segments is the number of marks - 1.
A ruler is complete if the set of all distances it can measure is {1,2,3,...,k} for some integer k>=1.
A ruler is perfect if it is complete and no complete ruler with the same length possesses less marks.
A ruler is optimal if it is perfect and no perfect ruler with the same number of segments has a greater length.
The 'empty ruler' with length n=0 is considered perfect and optimal.

			Rows begin:
[1],
[0,1],
[0,0,1],
[0,0,2,1],
[0,0,0,3,1],
[0,0,0,4,4,1],
[0,0,0,2,9,5,1],
[0,0,0,0,12,14,6,1],
[0,0,0,0,8,27,20,7,1],
...
a(19)=T(5,4)=4 counts the complete rulers with length 5 and 4 segments: {[0,2,3,4,5],[0,1,3,4,5],[0,1,2,4,5],[0,1,2,3,5]}

G. S. Bloom and S. W. Golomb, Numbered complete graphs, unusual rulers, and assorted applications. Theory and Applications of Graphs, Lecture Notes in Math. 642, (1978), 53-65.
R. K. Guy, Modular difference sets and error correcting codes. in: Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, chapter C10, pp. 181-183, 2004.
J. C. P. Miller, Difference bases: Three problems in additive number theory, pp. 299-322 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.

Fausto A. C. Cariboni, Rows n = 0..49, flattened (rows n = 0..39 from Hugo Pfoertner)
G. S. Bloom and S. W. Golomb, Applications of numbered undirected graphs, Proc. IEEE 65 (1977), 562-570.
Peter Luschny, Perfect and Optimal Rulers
Peter Luschny, Table of Counts
Peter Luschny, Perfect rulers
Eric Weisstein's World of Mathematics, Perfect Rulers
B. Wichmann, A note on restricted difference bases, J. Lond. Math. Soc. 38 (1963), 465-466.
Index entries for sequences related to perfect rulers

Row sums give A103295.
Column sums give A103296.
The first nonzero entries in the rows give A103300.
The last nonzero entries in the columns give A103299.
The row numbers of the last nonzero entries in the columns give A004137.
Cf. A103295 through A103301, A004137, A212661.

marks[n_, k_] := Module[{i}, i[0] = 0; iter = Sequence @@ Table[{i[j], i[j - 1] + 1, n - k + j - 1}, {j, 1, k}]; Table[Join[{0}, Array[i, k], {n}],
     iter // Evaluate] // Flatten[#, k - 1]&];
completeQ[ruler_List] := Range[ruler[[-1]]] == Sort[ Union[ Flatten[ Table[ ruler[[i]] - ruler[[j]], {i, 1, Length[ruler]}, {j, 1, i - 1}]]]];
rulers[n_, k_] := Select[marks[n, k - 1], completeQ];
T[n_, n_] = 1; T[, 0] = 0; T[n, k_] := Length[rulers[n, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Quiet (* Jean-François Alcover, Jul 05 2019 *)

def isComplete(R) :
    S = Set([])
    L = len(R)-1
    for i in range(L,0,-1) :
        for j in (1..i) :
            S = S.union(Set([R[i]-R[i-j]]))
    return len(S) == R[L]
def Partsum(T) :
    return [add([T[j] for j in range(i)]) for i in (0..len(T))]
def Ruler(L, S) :
    return map(Partsum, Compositions(L, length=S))
def CompleteRuler(L, S) :
    return tuple(filter(isComplete, Ruler(L, S)))
for n in (0..8):
    print([len(CompleteRuler(n,k)) for k in (0..n)]) # Peter Luschny, Jul 05 2019

Typo in data corrected by Jean-François Alcover, Jul 05 2019

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}]

A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 11, 16, 20, 24, 30, 43, 47, 62, 79, 94, 113, 143, 170, 211, 256, 307, 372, 449, 531, 648, 779, 926, 1100, 1323, 1562, 1864, 2190, 2595, 3053, 3611, 4242, 4977, 5834, 6825, 7973, 9344, 10844, 12641, 14699, 17072, 19822

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

A weak run-sum of a sequence is the sum of any consecutive constant subsequence. For example, the weak run-sums of (3,2,2,1) are {1,2,3,4}.

This is a kind of completeness property, cf. A126796.

			The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (211)   (311)    (321)     (3211)     (3221)
             (111)  (1111)  (2111)   (3111)    (4111)     (32111)
                            (11111)  (21111)   (22111)    (41111)
                                     (111111)  (31111)    (221111)
                                               (211111)   (311111)
                                               (1111111)  (2111111)
                                                          (11111111)

For parts instead of weak run-sums we have A000009.
For multiplicities instead of weak run-sums we have A317081.
If weak run-sums are distinct we have A353865, the completion of A353864.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A005811 counts runs in binary expansion.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A300273 ranks collapsible partitions, counted by A275870, comps A353860.
A353832 represents taking run-sums of a partition, compositions A353847.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353861 counts distinct weak run-sums of prime indices.
A353932 lists run-sums of standard compositions.
Rulers: A103295, A103300, A169942, A325768.
Complete: A002033, A325780, A126796, A276024, A325781, A188431, A353866.
Cf. A047967, A073093, A181819, A237685, A353844, A353867, A353930.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
wkrs[y_]:=Union[Total/@Select[msubs[y],SameQ@@#&]];
Table[Length[Select[IntegerPartitions[n],normQ[Rest[wkrs[#]]]&]],{n,0,15}]

\\ isok(p) tests the partition.
isok(p)={my(b=0, s=0, t=0); for(i=1, #p, if(p[i]<>t, t=p[i]; s=0); s += t; b = bitor(b, 1<<(s-1))); bitand(b,b+1)==0}
a(n) = {my(r=0); forpart(p=n, r+=isok(p)); r} \\ Andrew Howroyd, Jan 15 2024

a(31) onwards from Andrew Howroyd, Jan 15 2024

A325770 Number of distinct nonempty contiguous subsequences of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 20 2019

nonn

After a(1) = 0, first differs from A305611 at a(42) = 6, A305611(42) = 7.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(84) = 9 distinct nonempty contiguous subsequences of (4,2,1,1) are (1), (2), (4), (1,1), (2,1), (4,2), (2,1,1), (4,2,1), (4,2,1,1).

Cf. A002865, A103295, A112798, A124771, A276024, A325765, A325768, A325769, A335519, A335838.

Table[Length[Union[ReplaceList[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]],{_,s__,_}:>{s}]]],{n,30}]

a(n) = A335519(n) - 1.

Name corrected by Gus Wiseman, Jun 27 2020

A124770 Number of distinct nonempty subsequences for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 03 2020

			Composition number 11 is 2,1,1; the nonempty subsequences are 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 5.
The table starts:
  0
  1
  1 2
  1 3 3 3
  1 3 2 5 3 5 5 4
  1 3 3 5 3 5 5 7 3 5 5 8 5 8 7 5
From _Gus Wiseman_, Apr 03 2020: (Start)
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The STC-numbers of the distinct subsequences of the composition with STC-number k are given in column k below:
  1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2   1   2
        3     2  2  3     4  10  2   4   2   2   3       8   4   4   4
              5  6  7     9      3   12  6   3   7       17  18  3   20
                                 5       5   6   15              9
                                 11      13  14                  19
(End)

Alois P. Heinz, Rows n = 0..14, flattened

Row lengths are A011782.
Allowing empty subsequences gives A124771.
Dominates A333224, the version counting subsequence-sums instead of subsequences.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A108917, A143823, A325680.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>{s}]]],{n,0,100}] (* Gus Wiseman, Apr 03 2020 *)

a(n) = A124771(n) - 1. - Gus Wiseman, Apr 03 2020

A353390 Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 3, 2, 2, 8, 17, 26, 43, 77, 129, 210, 351, 569

Offset: 0

Gus Wiseman, May 15 2022

			The a(0) = 1 through a(9) = 8 compositions (empty columns indicated by dots):
  ()  (1)  .  .  (22)  (122)  (1122)  (11221)  (21122)  (333)
                       (221)  (1221)  (12211)  (22112)  (22113)
                              (2211)                    (22122)
                                                        (31122)
                                                        (121122)
                                                        (122112)
                                                        (211221)
                                                        (221121)
For example, the composition y = (2,2,3,3,1) has run-lengths (2,2,1), which form a (non-consecutive) subsequence, so y is counted under a(11).

The version for partitions is A325702.
The recursive version is A353391, ranked by A353431.
The consecutive case is A353392, ranked by A353432.
These compositions are ranked by A353402.
The reverse version is A353403.
The recursive consecutive version is A353430.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A047966 counts uniform partitions, compositions A329738.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223, partitions A108917.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-lengths, for runs A351013.
A353400 counts compositions with all run-lengths > 2.
Cf. A005811, A103295, A114901, A181591, A238279, A242882, A324572, A333755, A351017, A353401, A353426.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MemberQ[Subsets[#],Length/@Split[#]]&]],{n,0,15}]

A325685 Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 3, 5, 3, 9, 1, 9, 5, 7, 5, 11, 1, 13, 5, 9, 5, 13, 3, 13, 7, 9, 5, 17, 1, 17, 5, 9, 9, 15, 5, 15, 5, 13, 5, 21, 1, 17, 9, 9, 9, 17, 3, 21, 7, 13, 5, 17, 5, 21, 9, 13, 5, 21, 1, 21, 9, 11, 13, 19, 5, 17, 5, 17, 5, 29, 1, 21, 9, 9, 13, 17, 5, 25, 7, 17, 7

Offset: 0

Gus Wiseman, May 13 2019

nonn

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

Compare to the definition of perfect partitions (A002033).

			The distinct consecutive subsequences of (3,4,1,1) together with their sums are:
   1: {1}
   2: {1,1}
   3: {3}
   4: {4}
   5: {4,1}
   6: {4,1,1}
   7: {3,4}
   8: {3,4,1}
   9: {3,4,1,1}
Because the sums are all different and cover {1...9}, it follows that (3,4,1,1) is counted under a(9).
The a(1) = 1 through a(9) = 9 compositions:
  1   11   12    1111   113     132      1114      1133       1143
           21           122     231      1222      3311       1332
           111          221     111111   2221      11111111   2331
                        311              4111                 3411
                        11111            1111111              11115
                                                              12222
                                                              22221
                                                              51111
                                                              111111111

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100
Fausto A. C. Cariboni, All compositions that yield a(n) for n = 1..100, Feb 21 2022.

Cf. A000079, A002033, A103295, A126796, A143823, A169942, A325676, A325677, A325683, A325684.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Sort[Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]]==Range[n]&]],{n,0,15}]

a(21)-a(25) from Jinyuan Wang, Jun 26 2020

a(21)-a(25) corrected, a(26)-a(80) from Fausto A. C. Cariboni, Feb 21 2022

A325684 Number of minimal complete rulers of length n.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 12, 12, 24, 40, 46, 92, 133, 192, 308, 546, 710, 1108, 1754, 2726, 3878, 5928, 9260, 14238, 20502, 30812, 48378, 72232, 105744, 160308, 241592, 362348, 540362, 797750, 1183984, 1786714

Offset: 0

Gus Wiseman, May 13 2019

A complete ruler of length n is a subset of {0..n} containing 0 and n and such that the differences of distinct terms (up to sign) cover an initial interval of positive integers.

Also the number of maximal (most coarse) compositions of n whose consecutive subsequence-sums cover an initial interval of positive integers.

			The a(1) = 1 through a(7) = 12 rulers:
  {0,1}  {0,1,2}  {0,1,3}  {0,1,2,4}  {0,1,2,5}  {0,1,4,6}    {0,1,2,3,7}
                  {0,2,3}  {0,1,3,4}  {0,1,3,5}  {0,2,5,6}    {0,1,2,4,7}
                           {0,2,3,4}  {0,2,4,5}  {0,1,2,3,6}  {0,1,2,5,7}
                                      {0,3,4,5}  {0,1,3,5,6}  {0,1,3,5,7}
                                                 {0,3,4,5,6}  {0,1,3,6,7}
                                                              {0,1,4,5,7}
                                                              {0,1,4,6,7}
                                                              {0,2,3,6,7}
                                                              {0,2,4,6,7}
                                                              {0,2,5,6,7}
                                                              {0,3,5,6,7}
                                                              {0,4,5,6,7}
The a(1) = 1 through a(9) = 24 compositions:
  (1)  (11)  (12)  (112)  (113)  (132)   (1114)  (1133)   (1143)
             (21)  (121)  (122)  (231)   (1123)  (1241)   (1332)
                   (211)  (221)  (1113)  (1132)  (1322)   (2331)
                          (311)  (1221)  (1222)  (1412)   (3411)
                                 (3111)  (1231)  (1421)   (11115)
                                         (1312)  (2141)   (11124)
                                         (1321)  (2231)   (11142)
                                         (2131)  (3311)   (11241)
                                         (2221)  (11114)  (11322)
                                         (2311)  (11132)  (12141)
                                         (3211)  (23111)  (12222)
                                         (4111)  (41111)  (12231)
                                                          (12312)
                                                          (13221)
                                                          (14112)
                                                          (14121)
                                                          (14211)
                                                          (21141)
                                                          (21321)
                                                          (22221)
                                                          (22311)
                                                          (24111)
                                                          (42111)
                                                          (51111)

Cf. A000079, A103295, A126796, A143823, A169942, A325677, A325683, A325685.

fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Accumulate/@Select[Join@@Permutations/@IntegerPartitions[n],SubsetQ[ReplaceList[#,{_,s__,_}:>Plus[s]],Range[n]]&]]],{n,0,15}]

a(16)-a(36) from Fausto A. C. Cariboni, Feb 27 2022

A349976 Triangle read by rows, number of subsets S of [n] with |distset(S)| = k. T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 2, 3, 1, 5, 10, 4, 8, 4, 1, 6, 15, 6, 17, 10, 9, 1, 7, 21, 9, 31, 17, 25, 17, 1, 8, 28, 12, 51, 27, 47, 49, 33, 1, 9, 36, 16, 77, 43, 77, 97, 93, 63, 1, 10, 45, 20, 112, 62, 113, 169, 177, 187, 128

Offset: 0

Peter Luschny, Dec 09 2021

We use the notation [n] = {0, 1, ..., n-1}. If S is a subset of [n] then we define the distset of S (set of distances of S) as {|x - y|: x, y in S}.

For instance a subset S of [n] is a complete ruler (A103295) of length n - 1 if and only if distset(S) = [n].

			Triangle starts:
[n\k]  0, 1,  2,  3,  4,  5,  6,  7,  8,  9
-------------------------------------------
[ 0 ]  1;
[ 1 ]  1, 1;
[ 2 ]  1, 2,  1;
[ 3 ]  1, 3,  3,  1;
[ 4 ]  1, 4,  6,  2,  3;
[ 5 ]  1, 5, 10,  4,  8,  4;
[ 6 ]  1, 6, 15,  6, 17, 10,  9;
[ 7 ]  1, 7, 21,  9, 31, 17, 25, 17;
[ 8 ]  1, 8, 28, 12, 51, 27, 47, 49, 33;
[ 9 ]  1, 9, 36, 16, 77, 43, 77, 97, 93, 63;
.
Let S = {0, 3, 6, 7, 8}. Then S is a subset of [9] and distset(S) = [9].
For n = 7 the 9 subsets S with |distset(S)| = 3 are: {1, 2, 3}, {1, 3, 5}, {1, 4, 7}, {2, 3, 4}, {2, 4, 6}, {3, 4, 5}, {3, 5, 7}, {4, 5, 6}, {5, 6, 7}.

Fausto A. C. Cariboni, Rows n = 0..47, flattened (rows n = 0..36 from Peter Luschny)
Scott Harvey-Arnold, Steven J. Miller, and Fei Peng, Distribution of missing differences in diffsets, arXiv:2001.08931 [math.CO], 2020. Also in: Combinatorial and Additive Number Theory IV, Springer 2021. [Table 2 and Table 3, page 276 and 277 in the book.]
Peter Luschny, The first 37 rows, formatted as a triangular array.
Index entries for sequences related to perfect rulers.

Columns: A000012, A000027, A000217, A002620, A349975.
Cf. A000079 (row sums), A103295 (main diagonal), A349972 (subdiagonal).
Cf A349973, A349974, A350103.

distSetSize[s_] := Length @ Union[Map[Abs[Differences[#][[1]]] &, Union[Sort /@ Tuples[s, 2]]]]; T[n_, k_] := Count[Subsets[Range[0, n - 1]], ?(distSetSize[#] == k &)]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar, Dec 12 2021 *)

from collections import Counter
def DistsetLength(R) :
    S, L = Set([]), len(R)
    for r in R:
        for s in R:
            S = S.union(Set([abs(r - s)]))
    return len(S)
def A349976row(n):
    C = Counter(DistsetLength(s) for s in Subsets(n))
    return [C[k] for k in (0..n)]
for n in (0..9): print(A349976row(n))

cp's OEIS Frontend

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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A103294 Triangle T, read by rows: T(n,k) = number of complete rulers with length n and k segments (n >= 0, k >= 0).

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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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A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

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A325770 Number of distinct nonempty contiguous subsequences of the integer partition with Heinz number n.

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A124770 Number of distinct nonempty subsequences for compositions in standard order.

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A353390 Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).

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A325685 Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.

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