A325554 -id:A325554 - cp's OEIS frontend

A325545 Number of compositions of n with distinct differences.

1, 1, 2, 3, 7, 13, 17, 34, 59, 105, 166, 279, 442, 730, 1157, 1927, 3045, 4741, 7527, 11667, 18048, 27928, 43334, 65861, 101385, 153404, 232287, 347643, 523721, 780083, 1165331, 1725966, 2561625, 3773838, 5561577, 8151209, 11920717, 17364461, 25269939, 36635775

Offset: 0

Gus Wiseman, May 10 2019

nonn

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

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)   (3)   (4)    (5)     (6)
       (11)  (12)  (13)   (14)    (15)
             (21)  (22)   (23)    (24)
                   (31)   (32)    (33)
                   (112)  (41)    (42)
                   (121)  (113)   (51)
                   (211)  (122)   (114)
                          (131)   (132)
                          (212)   (141)
                          (221)   (213)
                          (311)   (231)
                          (1121)  (312)
                          (1211)  (411)
                                  (1131)
                                  (1221)
                                  (1311)
                                  (2112)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..70
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A059966, A070211, A175342, A242882, A325325, A325352, A325546, A325547, A325548, A325551, A325554.

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

More terms from Alois P. Heinz, May 11 2019

A325549 Number of necklace compositions of n with distinct circular differences.

Original entry on oeis.org

1, 1, 2, 3, 5, 4, 10, 16, 23, 34, 53, 66, 113, 164, 262, 380, 567, 821, 1217, 1778, 2702, 3919, 5760, 8520, 12375

Offset: 1

Gus Wiseman, May 10 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.

The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

			The a(1) = 1 through a(8) = 16 necklace compositions:
  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)
            (12)  (13)   (14)   (15)   (16)    (17)
                  (112)  (23)   (24)   (25)    (26)
                         (113)  (114)  (34)    (35)
                         (122)         (115)   (116)
                                       (124)   (125)
                                       (133)   (134)
                                       (142)   (143)
                                       (223)   (152)
                                       (1213)  (224)
                                               (233)
                                               (1124)
                                               (1142)
                                               (1214)
                                               (11213)
                                               (11312)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A034297, A059966, A070211, A318728, A318748, A320348, A325545, A325551, A325554, A325556.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&UnsameQ@@Append[Differences[#],First[#]-Last[#]]&]],{n,15}]

A325555 Number of necklace compositions of n with distinct differences up to sign.

Original entry on oeis.org

1, 2, 2, 4, 5, 6, 10, 15, 19, 24, 39, 49, 78, 106, 155, 207, 313, 430, 608, 867, 1239, 1670, 2313, 3220, 4483

Offset: 1

Gus Wiseman, May 11 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.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(8) = 15 necklace compositions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)
       (11)  (12)  (13)   (14)   (15)   (16)    (17)
                   (22)   (23)   (24)   (25)    (26)
                   (112)  (113)  (33)   (34)    (35)
                          (122)  (114)  (115)   (44)
                                 (132)  (124)   (116)
                                        (133)   (125)
                                        (142)   (134)
                                        (223)   (143)
                                        (1132)  (152)
                                                (224)
                                                (233)
                                                (1124)
                                                (1142)
                                                (1322)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A008965, A235998, A242882, A325325, A325352, A325404, A325468, A325552, A325554, A325556.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Abs[Differences[#]]&&neckQ[#]&]],{n,15}]

A325550 Number of necklace compositions of n with distinct multiplicities.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 11, 16, 18, 41, 86, 118, 273, 465, 731, 1432, 2791, 4063, 8429, 14761, 29465, 58654, 123799, 227419, 453229, 861909, 1697645, 3192807, 6315007, 11718879, 22795272, 42965245, 83615516, 156215020, 306561088, 587300503, 1140650287, 2203107028

Offset: 1

Gus Wiseman, May 10 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.

			The a(1) = 1 through a(8) = 16 necklace compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (113)    (33)      (115)      (44)
                    (112)   (122)    (114)     (133)      (116)
                    (1111)  (1112)   (222)     (223)      (224)
                            (11111)  (1113)    (1114)     (233)
                                     (11112)   (1222)     (1115)
                                     (111111)  (11113)    (2222)
                                               (11122)    (11114)
                                               (11212)    (11222)
                                               (111112)   (12122)
                                               (1111111)  (111113)
                                                          (111122)
                                                          (111212)
                                                          (112112)
                                                          (1111112)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000079, A000740, A008965, A059966, A098504, A098859, A242882, A242887, A325549, A325554.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&UnsameQ@@Length/@Split[Sort[#]]&]],{n,15}]

b(n)={((r,k,b,w)->if(!k||!r, if(r,0,(w-1)!), sum(m=0, r\k, if(!m || !bittest(b,m), self()(r-k*m, k-1, bitor(b,1<Andrew Howroyd, Aug 31 2019

a(n) = Sum_{d|n} phi(d)*(Sum_{k=1..n/d} A242887(n/d, k)/k)/d. - Andrew Howroyd, Aug 31 2019

Terms a(26) and beyond from Andrew Howroyd, Aug 31 2019

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A325545 Number of compositions of n with distinct differences.

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A325549 Number of necklace compositions of n with distinct circular differences.

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A325555 Number of necklace compositions of n with distinct differences up to sign.

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A325550 Number of necklace compositions of n with distinct multiplicities.

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