A169942 -id:A169942 - cp's OEIS frontend

A325678 Maximum length of a composition of n such that every restriction to a subinterval has a different sum.

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

Offset: 0

Gus Wiseman, May 13 2019

nonn

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

Also the maximum number of nonzero marks on a Golomb ruler of length n.

Run-lengths are A270813.
Cf. A000079, A007318, A103295, A108917, A143823, A169942.
Cf. A325466, A325676, A325677, A325679, A325681, A325683, A325687.

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

a(n) + 1 = A143824(n + 1).

A325779 Heinz numbers of integer partitions for which every restriction to a subinterval has a different sum.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107

Offset: 1

Gus Wiseman, May 20 2019

nonn

First differs from A301899 in having 462.

The enumeration of these partitions by sum is given by A325768.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   49: {4,4}
   50: {1,3,3}

A subsequence of A005117.

Cf. A000041, A002033, A056239, A103300, A112798, A143823, A169942, A299702, A301899, A325676, A325768, A325769, A325770, A325778.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@ReplaceList[primeMS[#],{_,s__,_}:>Plus[s]]&]

A353430 Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 16, 22, 25, 37, 37, 45

Offset: 0

Gus Wiseman, May 16 2022

			The a(n) compositions for selected n (A..E = 10..14):
  n=4:  n=6:    n=9:      n=10:     n=12:     n=14:
-----------------------------------------------------------
  (4)   (6)     (9)       (A)       (C)       (E)
  (22)  (1122)  (333)     (2233)    (2244)    (2255)
        (2211)  (121122)  (3322)    (4422)    (5522)
                (221121)  (131122)  (151122)  (171122)
                          (221131)  (221124)  (221126)
                                    (221142)  (221135)
                                    (221151)  (221153)
                                    (241122)  (221162)
                                    (421122)  (221171)
                                              (261122)
                                              (351122)
                                              (531122)
                                              (621122)
                                              (122121122)
                                              (221121221)

Non-recursive non-consecutive version: counted by A353390, ranked by A353402, reverse A353403, partitions A325702.
Non-consecutive version: A353391, ranked by A353431, partitions A353426.
Non-recursive version: A353392, ranked by A353432.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A114901 counts compositions with no runs of length 1.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A329738 counts uniform compositions, partitions A047966.
A329739 counts compositions with all distinct run-lengths.
Cf. A005811, A032020, A103295, A114640, A165413, A242882, A325705, A333755, A351013, A353400, A353401.

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]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yoyQ]],{n,0,15}]

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

A036501 Number of inequivalent Golomb rulers with n marks and shortest length.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

N. J. A. Sloane

From Gus Wiseman, May 31 2019: (Start)
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. For example, the a(2) = 1 through a(8) = 1 Golomb rulers are:
{0,1}
{0,1,3}
{0,1,4,6}
{0,1,4,9,11}
{0,2,7,8,11}
{0,1,4,10,12,17}
{0,1,4,10,15,17}
{0,1,8,11,13,17}
{0,1,8,12,14,17}
{0,1,4,10,18,23,25}
{0,1,7,11,20,23,25}
{0,2,3,10,16,21,25}
{0,2,7,13,21,22,25}
{0,1,11,16,19,23,25}
{0,1,4,9,15,22,32,34}
Also half the number of length-(n - 1) compositions of A003022(n) such that every consecutive subsequence has a different sum. For example, the a(2) = 1 through a(8) = 1 compositions are (A = 10):
(1)
(1,2)
(1,3,2)
(1,3,5,2)
(2,5,1,3)
(1,3,6,2,5)
(1,3,6,5,2)
(1,7,3,2,4)
(1,7,4,2,3)
(1,3,6,8,5,2)
(1,6,4,9,3,2)
(2,1,7,6,5,4)
(2,5,6,8,1,3)
(1,A,5,3,4,2)
(1,3,5,6,7,A,2)
(End)

Anonymous, In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers
Distributed.Net, Project OGR
J. B. Shearer, Table of known optimal Golomb rulers
Eric Weisstein's World of Mathematics, Golomb Ruler
Index entries for sequences related to Golomb rulers

Cf. A003022, A039953, A054578, A108917, A143823, A169942, A270813, A325677, A325683.

A325682 Number of necklace compositions of n such that every distinct circular subsequence has a different sum.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 9, 13, 12, 17, 21, 28, 26, 49, 46, 74, 68, 113, 107, 176, 144, 255, 235, 375

Offset: 1

Gus Wiseman, May 13 2019

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive.

			The a(1) = 1 through a(8) = 13 necklace compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (1111)  (11111)  (33)      (34)       (35)
                                     (222)     (124)      (44)
                                     (111111)  (142)      (125)
                                               (1111111)  (152)
                                                          (2222)
                                                          (11111111)

Cf. A000079, A000740, A008965, A059966, A108917, A143823, A169942, A276024.

Cf. A325676, A325680, A325685, A325687.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&UnsameQ@@Total/@subalt[#]&]],{n,20}]

a(21)-a(25) from Robert Price, Jun 19 2021

A325763 Heinz numbers of integer partitions whose consecutive subsequence-sums cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 96, 100, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 216, 224, 240, 256, 280, 288, 300, 320, 324, 336, 352, 360, 384, 392, 400, 416, 432, 448, 480, 486, 500, 504, 512

Offset: 1

Gus Wiseman, May 19 2019

nonn

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

The enumeration of these partitions by sum appears to be A002865.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    24: {1,1,1,2}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}

Cf. A002033, A103295, A103300, A169942, A325676, A325685, A325764, A325765.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Range[Total[primeMS[#]]]==Union[ReplaceList[primeMS[#],{_,s__,_}:>Plus[s]]]&]

A334967 Numbers k such that the every subsequence (not necessarily contiguous) of the k-th composition in standard order (A066099) 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, 42, 48, 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, 134

Offset: 1

Gus Wiseman, Jun 02 2020

nonn

First differs from A333223 in lacking 41.

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:
   0: ()           18: (3,2)          48: (1,5)
   1: (1)          19: (3,1,1)        56: (1,1,4)
   2: (2)          20: (2,3)          63: (1,1,1,1,1,1)
   3: (1,1)        21: (2,2,1)        64: (7)
   4: (3)          24: (1,4)          65: (6,1)
   5: (2,1)        26: (1,2,2)        66: (5,2)
   6: (1,2)        28: (1,1,3)        67: (5,1,1)
   7: (1,1,1)      31: (1,1,1,1,1)    68: (4,3)
   8: (4)          32: (6)            69: (4,2,1)
   9: (3,1)        33: (5,1)          70: (4,1,2)
  10: (2,2)        34: (4,2)          71: (4,1,1,1)
  12: (1,3)        35: (4,1,1)        72: (3,4)
  15: (1,1,1,1)    36: (3,3)          73: (3,3,1)
  16: (5)          40: (2,4)          74: (3,2,2)
  17: (4,1)        42: (2,2,2)        80: (2,5)

These compositions are counted by A334268.
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.
The case of partitions is counted by A325769 and ranked by A325778.
Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
Number of (not necessarily contiguous) subsequences is A334299.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A124771, A325770, A334300, A334967.

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

A325681 Number of necklace compositions of n such that every restriction to a circular subinterval has a different sum.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 6, 11, 9, 16, 16, 27, 23, 46, 42, 73, 63, 112, 102, 173, 141, 254, 228, 373, 313, 614, 500, 855, 709, 1252, 1074, 1827, 1457, 2470, 2260, 3559, 2905, 5044, 4294, 6997, 5623, 9752, 8422, 13741, 10913, 18562, 15912, 25213, 20569, 35146, 29286, 46307, 38241, 61396

Offset: 1

Gus Wiseman, May 13 2019

nonn

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

A circular subinterval is a sequence of consecutive indices where the first and last indices are also considered consecutive.

			The a(1) = 1 through a(10) = 9 necklace compositions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)
            (12)  (13)  (14)  (15)  (16)   (17)   (18)   (19)
                        (23)  (24)  (25)   (26)   (27)   (28)
                                    (34)   (35)   (36)   (37)
                                    (124)  (125)  (45)   (46)
                                    (142)  (152)  (126)  (127)
                                                  (135)  (136)
                                                  (153)  (163)
                                                  (162)  (172)
                                                  (234)
                                                  (243)

Andrew Howroyd, Table of n, a(n) for n = 1..80

Cf. A000079, A000740, A008965, A108917, A143823, A169942, A276024.

Cf. A325676, A325677, A325678, A325679, A325682, A325683, A325687.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
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],neckQ[#]&&UnsameQ@@Total/@suball[#]&]],{n,15}]

a(n)={
   my(recurse(k,r,b,w)=
      if(k >= n, 1/r,
         b+=1<Andrew Howroyd, Mar 25 2025

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

A325765 Number of integer partitions of n with a unique consecutive subsequence summing to every positive integer from 1 to n.

Original entry on oeis.org

1, 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, 7, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3

Offset: 0

Gus Wiseman, May 20 2019

After a(0) = 1, same as A032741(n + 1) (number of proper divisors of n + 1).

The Heinz numbers of these partitions are given by A325764.

			The a(1) = 1 through a(13) = 3 partitions:
  (1)  (11)  (21)   (1111)  (221)    (111111)  (2221)     (3311)
             (111)          (311)              (4111)     (11111111)
                            (11111)            (1111111)
.
  (22221)      (1111111111)  (33311)        (111111111111)  (2222221)
  (51111)                    (44111)                        (7111111)
  (111111111)                (222221)                       (1111111111111)
                             (611111)
                             (11111111111)

Cf. A000041, A002033, A103295, A103300, A143823, A169942, A325676, A325683, A325768, A325769, A325770.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n],normQ[Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]]&&UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,0,20}]

cp's OEIS Frontend

A325678 Maximum length of a composition of n such that every restriction to a subinterval has a different sum.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A325779 Heinz numbers of integer partitions for which every restriction to a subinterval has a different sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353430 Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A036501 Number of inequivalent Golomb rulers with n marks and shortest length.

Views

Author

Keywords

Comments

Links

Crossrefs

A325682 Number of necklace compositions of n such that every distinct circular subsequence has a different sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A325763 Heinz numbers of integer partitions whose consecutive subsequence-sums cover an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A334967 Numbers k such that the every subsequence (not necessarily contiguous) of the k-th composition in standard order (A066099) has a different sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325681 Number of necklace compositions of n such that every restriction to a circular subinterval has a different sum.

Views

Author

Keywords

Comments

Examples

Links