A325682 -id:A325682

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A325676

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

+10
57

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

Compare to the definition of knapsack partitions (A108917).

LINKS

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

EXAMPLE

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)

MATHEMATICA

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

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 13 2019

EXTENSIONS

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

STATUS

approved

A325680

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

+10
13

1, 1, 2, 4, 5, 6, 8, 14, 16, 29, 24, 42, 46, 78, 66, 146, 133, 242, 208, 386, 352, 620, 494, 948, 842, 1447

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

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

LINKS

Table of n, a(n) for n=0..25.

EXAMPLE

The a(1) = 1 through a(8) = 16 compositions:

(1) (2) (3) (4) (5) (6) (7) (8)

(11) (12) (13) (14) (15) (16) (17)

(21) (22) (23) (24) (25) (26)

(111) (31) (32) (33) (34) (35)

(1111) (41) (42) (43) (44)

(11111) (51) (52) (53)

(222) (61) (62)

(111111) (124) (71)

(142) (125)

(214) (152)

(241) (215)

(412) (251)

(421) (512)

(1111111) (521)

(2222)

(11111111)

MATHEMATICA

subalt[q_]:=Union[ReplaceList[q, {___, s__, ___}:>{s}], DeleteCases[ReplaceList[q, {t___, __, u___}:>{u, t}], {}]];

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

CROSSREFS

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

Cf. A325545, A325676, A325682, A325685, A325687, A325688.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 13 2019

EXTENSIONS

a(18)-a(25) from Robert Price, Jun 19 2021

STATUS

approved

A354580

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

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

EXAMPLE

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)

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 13 2022

EXTENSIONS

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

STATUS

approved

A325786

Number of complete necklace compositions of n.

+10
5

1, 1, 2, 2, 4, 7, 12, 19, 41, 71, 141, 255, 509, 924, 1882, 3395, 6838, 12715, 25233, 47049

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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. A necklace composition of n is complete if every positive integer from 1 to n is the sum of some circular subsequence.

LINKS

Table of n, a(n) for n=1..20.

EXAMPLE

The a(1) = 1 through a(8) = 19 necklace compositions:

(1) (11) (12) (112) (113) (123) (124) (1124)

(111) (1111) (122) (132) (142) (1133)

(1112) (1113) (1114) (1142)

(11111) (1122) (1123) (1214)

(1212) (1132) (1223)

(11112) (1213) (1322)

(111111) (1222) (11114)

(11113) (11123)

(11122) (11132)

(11212) (11213)

(111112) (11222)

(1111111) (11312)

(12122)

(111113)

(111122)

(111212)

(112112)

(1111112)

(11111111)

MATHEMATICA

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[#]&&Union[Total/@subalt[#]]==Range[n]&]], {n, 15}]

CROSSREFS

Cf. A000740, A002033, A008965, A103295, A108917, A126796, A276024, A325549, A325682, A325781, A325788, A325789, A325791.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 22 2019

STATUS

approved

A325787

Number of perfect strict necklace compositions of n.

+10
5

1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

COMMENTS

A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is perfect if every positive integer from 1 to n is the sum of exactly one distinct circular subsequence. For example, the composition (1,2,6,4) is perfect because it has the following circular subsequences and sums:

1: (1)

2: (2)

3: (1,2)

4: (4)

5: (4,1)

6: (6)

7: (4,1,2)

8: (2,6)

9: (1,2,6)

10: (6,4)

11: (6,4,1)

12: (2,6,4)

13: (1,2,6,4)

a(n) > 0 iff n = A002061(k) = A004136(k) for some k. - Bert Dobbelaere, Nov 11 2020

LINKS

Bert Dobbelaere, Table of n, a(n) for n = 1..306

EXAMPLE

The a(1) = 1 through a(31) = 10 perfect strict necklace compositions (empty columns not shown):

(1) (1,2) (1,2,4) (1,2,6,4) (1,3,10,2,5) (1,10,8,7,2,3)

(1,4,2) (1,3,2,7) (1,5,2,10,3) (1,13,6,4,5,2)

(1,4,6,2) (1,14,4,2,3,7)

(1,7,2,3) (1,14,5,2,6,3)

(1,2,5,4,6,13)

(1,2,7,4,12,5)

(1,3,2,7,8,10)

(1,3,6,2,5,14)

(1,5,12,4,7,2)

(1,7,3,2,4,14)

From Bert Dobbelaere, Nov 11 2020: (Start)

Compositions matching nonzero terms from a(57) to a(273), up to symmetry.

a(57) = 12:

(1,2,10,19,4,7,9,5)

(1,3,5,11,2,12,17,6)

(1,3,8,2,16,7,15,5)

(1,4,2,10,18,3,11,8)

(1,4,22,7,3,6,2,12)

(1,6,12,4,21,3,2,8)

a(73) = 8:

(1,2,4,8,16,5,18,9,10)

(1,4,7,6,3,28,2,8,14)

(1,6,4,24,13,3,2,12,8)

(1,11,8,6,4,3,2,22,16)

a(91) = 12:

(1,2,6,18,22,7,5,16,4,10)

(1,3,9,11,6,8,2,5,28,18)

(1,4,2,20,8,9,23,10,3,11)

(1,4,3,10,2,9,14,16,6,26)

(1,5,4,13,3,8,7,12,2,36)

(1,6,9,11,29,4,8,2,3,18)

a(133) = 36:

(1,2,9,8,14,4,43,7,6,10,5,24)

(1,2,12,31,25,4,9,10,7,11,16,5)

(1,2,14,4,37,7,8,27,5,6,13,9)

(1,2,14,12,32,19,6,5,4,18,13,7)

(1,3,8,9,5,19,23,16,13,2,28,6)

(1,3,12,34,21,2,8,9,5,6,7,25)

(1,3,23,24,6,22,10,11,18,2,5,8)

(1,4,7,3,16,2,6,17,20,9,13,35)

(1,4,16,3,15,10,12,14,17,33,2,6)

(1,4,19,20,27,3,6,25,7,8,2,11)

(1,4,20,3,40,10,9,2,15,16,6,7)

(1,5,12,21,29,11,3,16,4,22,2,7)

(1,7,13,12,3,11,5,18,4,2,48,9)

(1,8,10,5,7,21,4,2,11,3,26,35)

(1,14,3,2,4,7,21,8,25,10,12,26)

(1,14,10,20,7,6,3,2,17,4,8,41)

(1,15,5,3,25,2,7,4,6,12,14,39)

(1,22,14,20,5,13,8,3,4,2,10,31)

a(183) = 40:

(1,2,13,7,5,14,34,6,4,33,18,17,21,8)

(1,2,21,17,11,5,9,4,26,6,47,15,12,7)

(1,2,28,14,5,6,9,12,48,18,4,13,16,7)

(1,3,5,6,25,32,23,10,18,2,17,7,22,12)

(1,3,12,7,20,14,44,6,5,24,2,28,8,9)

(1,3,24,6,12,14,11,55,7,2,8,5,16,19)

(1,4,6,31,3,13,2,7,14,12,17,46,8,19)

(1,4,8,52,3,25,18,2,9,24,6,10,7,14)

(1,4,20,2,12,3,6,7,33,11,8,10,35,31)

(1,5,2,24,15,29,14,21,13,4,33,3,9,10)

(1,5,23,27,42,3,4,11,2,19,12,10,16,8)

(1,6,8,22,4,5,33,21,3,20,32,16,2,10)

(1,8,3,10,23,5,56,4,2,14,15,17,7,18)

(1,8,21,45,6,7,11,17,3,2,10,4,23,25)

(1,9,5,40,3,4,21,35,16,18,2,6,11,12)

(1,9,14,26,4,2,11,5,3,12,27,34,7,28)

(1,9,21,25,3,4,8,5,6,16,2,36,14,33)

(1,10,22,34,27,12,3,4,2,14,24,5,8,17)

(1,10,48,9,19,4,8,6,7,17,3,2,34,15)

(1,12,48,6,2,38,3,22,7,10,11,5,4,14)

a(273) = 12:

(1,2,4,8,16,32,27,26,11,9,45,13,10,29,5,17,18)

(1,3,12,10,31,7,27,2,6,5,19,20,62,14,9,28,17)

(1,7,3,15,33,5,24,68,2,14,6,17,4,9,19,12,34)

(1,7,12,44,25,41,9,17,4,6,22,33,13,2,3,11,23)

(1,7,31,2,11,3,9,36,17,4,22,6,18,72,5,10,19)

(1,21,11,50,39,13,6,4,14,16,25,26,3,2,7,8,27)

(End)

MATHEMATICA

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/@Select[IntegerPartitions[n], UnsameQ@@#&], neckQ[#]&&Sort[Total/@subalt[#]]==Range[n]&]], {n, 30}]

CROSSREFS

Cf. A002033, A002061, A004136, A008965, A032153, A103300, A126796, A325680, A325682, A325780, A325782, A325786, A325788, A325789.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 22 2019

EXTENSIONS

More terms from Bert Dobbelaere, Nov 11 2020

STATUS

approved

A325681

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

+10
3

1, 1, 2, 2, 3, 3, 6, 6, 11, 9, 16, 16, 27, 23, 46, 42, 73, 63, 112, 102

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..20.

EXAMPLE

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)

MATHEMATICA

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}]

CROSSREFS

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

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

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 13 2019

STATUS

approved

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