A325680 -id:A325680 - cp's OEIS frontend

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

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

A325787 Number of perfect strict necklace compositions of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 22 2019

nonn

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)
(2)
(1,2)
(4)
(4,1)
(6)
(4,1,2)
(2,6)
(1,2,6)
(6,4)
(6,4,1)
(2,6,4)
(1,2,6,4)
a(n) > 0 iff n = A002061(k) = A004136(k) for some k. - Bert Dobbelaere, Nov 11 2020

			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)

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

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

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}]

More terms from Bert Dobbelaere, Nov 11 2020

A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 5, 10, 10, 24, 24, 43, 42, 88, 72, 136, 122, 242, 213, 392, 320, 630, 490, 916, 742, 1432, 1160, 1955, 1604, 2826, 2310, 3850, 2888, 5416, 4426, 7332, 5814, 10046, 7983, 12946, 10236, 17780, 14100, 22674, 17582, 30232, 23674, 37522, 29426, 49832

Offset: 0

Gus Wiseman, Jun 02 2020

nonn

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

The contiguous case is A325676.

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,1,1)  (4,1)        (4,2)
                                  (1,1,3)      (5,1)
                                  (1,2,2)      (1,1,4)
                                  (2,2,1)      (2,2,2)
                                  (3,1,1)      (4,1,1)
                                  (1,1,1,1,1)  (1,1,1,1,1,1)

These compositions are ranked by A334967.
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 partitions are counted by A108917 and A325592 and ranked by A299702, while the strict case is 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.
Cf. A003022, A029931, A103295, A143823, A325680, A333224.

b:= proc(n, s) option remember; `if`(n=0, 1, add((h->
      `if`(nops(h)=nops(map(l-> add(i, i=l), h)),
       b(n-j, h), 0))({s[], map(l-> [l[], j], s)[]}), j=1..n))
    end:
a:= n-> b(n, {[]}):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 03 2020

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

a(18)-a(47) from Alois P. Heinz, Jun 03 2020

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

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A325787 Number of perfect strict necklace compositions of n.

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A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

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