A325680 -id:A325680 - cp's OEIS frontend

A325676 Number of compositions of n such that every distinct consecutive subsequence has a different sum.

1, 1, 2, 4, 5, 10, 12, 24, 26, 47, 50, 96, 104, 172, 188, 322, 335, 552, 590, 938, 1002, 1612, 1648, 2586, 2862, 4131, 4418, 6718, 7122, 10332, 11166, 15930, 17446, 24834, 26166, 37146, 41087, 55732, 59592, 84068, 89740, 122106, 133070, 177876, 194024, 262840, 278626

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 knapsack partitions (A108917).

			The distinct consecutive subsequences of (1,4,4,3) together with their sums are:
   1: {1}
   3: {3}
   4: {4}
   5: {1,4}
   7: {4,3}
   8: {4,4}
   9: {1,4,4}
  11: {4,4,3}
  12: {1,4,4,3}
Because the sums are all different, (1,4,4,3) is counted under a(12).
The a(1) = 1 through a(6) = 12 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (1111)  (41)     (42)
                            (113)    (51)
                            (122)    (114)
                            (221)    (132)
                            (311)    (222)
                            (11111)  (231)
                                     (411)
                                     (111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100

Cf. A000079, A103295, A108917, A169942, A235998, A321143.

Cf. A325466, A325545, A325680, A325682, A325685, A325687, A325688.

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

a(21)-a(22) from Jinyuan Wang, Jun 20 2020
a(23)-a(25) from Robert Price, Jun 19 2021
a(26)-a(46) from Fausto A. C. Cariboni, Feb 10 2022

A333224 Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 4, 3, 4, 4, 4, 1, 3, 3, 5, 3, 5, 4, 5, 3, 4, 5, 5, 5, 5, 5, 5, 1, 3, 3, 5, 2, 5, 5, 6, 3, 6, 3, 6, 5, 6, 5, 6, 3, 4, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 1, 3, 3, 5, 3, 6, 6, 7, 3, 5, 5, 7, 4, 6, 6, 7, 3, 6, 4, 7, 5, 7, 6

Offset: 0

Gus Wiseman, Mar 18 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 composition (4,3,1,2) has positive subsequence-sums 1, 2, 3, 4, 6, 7, 8, 10, so a(550) = 8.

Dominated by A124770.
Compositions where every 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 partitions are counted by A108917 and A325592 and ranked by A299702.
Strict knapsack partitions are counted by A275972 and ranked by A059519 and A301899.
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.
Allowing empty subsequences gives A333257.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A124767, A143823, A295235, A325680, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>Plus[s]]]],{n,0,100}]

a(n) = A333257(n) - 1.

A124771 Number of distinct subsequences for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.
From Vladimir Shevelev, Dec 18 2013: (Start)
Every number in binary is a concatenation of parts of the form 10...0 with k>=0 zeros. For example, 5=(10)(1), 11=(10)(1)(1), 7=(1)(1)(1). We call d>0 a c-divisor of m, if d consists of some consecutive parts of m taking from the left to the right. Note that, to d=0 corresponds an empty set of parts. So it is natural to consider 0 as a c-divisor of every m. For example, 5=(10)(1) is a divisor of 23=(10)(1)(1)(1). Analogously, 1,2,3,7,11,23 are c-divisors of 23. But 6=(1)(10) is not a c-divisor of 23.
One can prove a one-to-one correspondence between distinct subsequences for  composition no. n in standard order and c-divisors of n. So, the sequence lists also numbers of c-divisors of nonnegative integers.
(End)
These are contiguous subsequences, or restrictions to a subinterval. The case for all subsequences is A334299. - Gus Wiseman, Jun 02 2020

			Composition number 11 is 2,1,1; the subsequences are (empty); 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 6.
The table starts:
1
2
1 2
1 3 3 3
Let n=11=(10)(1)(1). We have the following c-divisors of 11: 0,1,2,3,5,11. Thus a(11)=6. Note, that 3=(1)(1) is not a c-divisor of 13=(1)(10)(1) since, although it contains parts of 3=(1)(1), but in non-consecutive order. The c-divisors of 13 are 0,1,2,5,6,13. So, a(13)=6.
From _Gus Wiseman_, Jun 01 2020: (Start)
The c-divisors of n are given in column n below:
  0  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0
     1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2
           3     2  2  3     4  10  2   4   2   2   3       8   4
                 5  6  7     9      3   12  5   3   7       17  18
                                    5       6   6   15
                                    11      13  14
(End)

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

Cf. A000005, A011782 (row lengths), A066099, A114994, A233249, A233312.
Not allowing empty subsequences gives A124770.
Dominates A333257.
The case for not just contiguous subsequences is A334299.
Positions of first appearances are A335279.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Cf. A000120, A003022, A029931, A070939, A108917, A124767, A325680, A325770, A333224, A334967, A334968.

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, Jun 01 2020 *)

a(n) = A124770(n) + 1.
From Vladimir Shevelev, Dec 18 2013: (Start)
a(2^n) = 2. Note that in concatenation representations of integers in binary, numbers {2^k}, k>=0, play the role of primes. So the formula is an analog of A000005(prime(n))=2.
a(2^n-1) = n+1; for n>=2, a(2^n+1) = 4.
For c-equivalent numbers n_1 and n_2 (i.e., differed only by order of parts) we have a(n_1) = a(n_2). For example, a(24)=a(17)=4. If the canonical representation of n is n=(1)^k_1[*](10)^k_2[*](100)^k_3[*]... , where [*] denotes operation of concatenation (cf. A233569), then a(n)<=(k_1+1)*(k_2+1)*...
(End)

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}]]&]

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]]&]

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}]

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

A328609 Number of compositions of n whose circularly adjacent parts are relatively prime.

Original entry on oeis.org

1, 1, 1, 3, 6, 12, 23, 42, 81, 150, 284, 534, 1004, 1882, 3532, 6630, 12459, 23406, 43951, 82537, 154998, 291087, 546673, 1026686, 1928117, 3621016, 6800299, 12771085, 23984328, 45042958, 84591338, 158863806, 298348612, 560303341, 1052258401, 1976157509

Offset: 0

Gus Wiseman, Oct 26 2019

nonn

Circularity means the last part is followed by the first.

			The a(1) = 1 through a(6) = 23 compositions:
  (1)  (11)  (12)   (13)    (14)     (15)
             (21)   (31)    (23)     (51)
             (111)  (112)   (32)     (114)
                    (121)   (41)     (123)
                    (211)   (113)    (132)
                    (1111)  (131)    (141)
                            (311)    (213)
                            (1112)   (231)
                            (1121)   (312)
                            (1211)   (321)
                            (2111)   (411)
                            (11111)  (1113)
                                     (1131)
                                     (1212)
                                     (1311)
                                     (2121)
                                     (3111)
                                     (11112)
                                     (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The necklace version is A328597 or A318728 (with singletons).
The aperiodic version is A328670.
The Lyndon word version is A318745.
The version with singletons is A318748.
The non-circular version is A167606.
Relatively prime compositions are A000740.
Compositions with no part circularly followed by a divisor are A328598.
Cf. A008965, A178470, A318726, A325680, A328172, A328335, A328599, A328600.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@CoprimeQ@@@Partition[#,2,1,1]&]],{n,10}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={concat([1], sum(k=1, n, b(n, k, (i, j)->gcd(i, j)==1)))} \\ Andrew Howroyd, Nov 01 2019

a(n > 1) = A318748(n) - 1.

A325679 Number of compositions of n such that every restriction to a circular subinterval has a different sum.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 5, 13, 13, 27, 21, 41, 41, 77, 63, 143, 129, 241, 203, 385, 347, 617, 491, 947, 835, 1445, 1185, 2511, 1991, 3585, 2915, 5411, 4569, 8063, 6321, 11131, 10133, 16465, 13207, 23817, 20133, 33929, 26663, 48357, 41363, 69605, 54363, 95727, 81183, 132257, 106581

Offset: 0

Gus Wiseman, May 13 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.
A circular subinterval is a sequence of consecutive indices where the first and last indices are also considered consecutive.
For n > 0, a(n) is the number of subsets of Z_n which contain 0 and such that every ordered pair of distinct elements has a different difference (modulo n). The elements of a subset correspond with the partial sums of a composition. For example, when n = 8 the subset {0,2,7} corresponds with the composition (251). - Andrew Howroyd, Mar 24 2025

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)
            (12)  (13)  (14)  (15)  (16)   (17)
            (21)  (31)  (23)  (24)  (25)   (26)
                        (32)  (42)  (34)   (35)
                        (41)  (51)  (43)   (53)
                                    (52)   (62)
                                    (61)   (71)
                                    (124)  (125)
                                    (142)  (152)
                                    (214)  (215)
                                    (241)  (251)
                                    (412)  (512)
                                    (421)  (521)

Andrew Howroyd, Table of n, a(n) for n = 0..80

Cf. A000079, A008965, A108917, A143823, A169942, A276024.

Cf. A325545, A325676, A325677, A325678, A325680, A325681, A325687, A382399.

suball[q_]:=Join[Take[q,#]&/@Select[Tuples[Range[Length[q]],2],OrderedQ],Drop[q,#]&/@Select[Tuples[Range[2,Length[q]-1],2],OrderedQ]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@suball[#]&]],{n,0,15}]

a(n)={
   my(recurse(k,b,w)=
      if(k >= n, 1,
         b+=1<Andrew Howroyd, Mar 24 2025

a(21) onwards from Andrew Howroyd, Mar 24 2025

A354580 Number of rucksack compositions of n: every distinct partial run has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 22, 39, 68, 125, 227, 402, 710, 1280, 2281, 4040, 7196, 12780, 22623, 40136, 71121, 125863, 222616, 393305, 695059, 1227990, 2167059, 3823029, 6743268, 11889431, 20955548, 36920415, 65030404, 114519168, 201612634, 354849227

Offset: 0

Gus Wiseman, Jun 13 2022

nonn

We define a partial run of a sequence to be any contiguous constant subsequence. The term rucksack is short for run-knapsack.

			The a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)    (3)      (4)        (5)
           (1,1)  (1,2)    (1,3)      (1,4)
                  (2,1)    (2,2)      (2,3)
                  (1,1,1)  (3,1)      (3,2)
                           (1,2,1)    (4,1)
                           (1,1,1,1)  (1,1,3)
                                      (1,2,2)
                                      (1,3,1)
                                      (2,1,2)
                                      (2,2,1)
                                      (3,1,1)
                                      (1,1,1,1,1)

Max Alekseyev, Table of n, a(n) for n = 0..65

The knapsack version is A325676, ranked by A333223.
The non-partial version for partitions is A353837, ranked by A353838 (complement A353839).
The non-partial version is A353850, ranked by A353852.
The version for partitions is A353864, ranked by A353866.
The complete version for partitions is A353865, ranked by A353867.
These compositions are ranked by A354581.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A108917 counts knapsack partitions, ranked by A299702, strict A275972.
A238279 and A333755 count compositions by number of runs.
A275870 counts collapsible partitions, ranked by A300273.
A353836 counts partitions by number of distinct run-sums.
A353847 is the composition run-sum transformation.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions, ranked by A354908.
Cf. A143823, A169942, A242882, A325545, A325680, A325682, A325685, A325687, A329739, A351017, A353849.

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

Terms a(16) onward from Max Alekseyev, Sep 10 2023

cp's OEIS Frontend

A325676 Number of compositions of n such that every distinct consecutive subsequence has a different sum.

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A333224 Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.

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A124771 Number of distinct subsequences for compositions in standard order.

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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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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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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.

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

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A328609 Number of compositions of n whose circularly adjacent parts are relatively prime.

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