A325551 -id:A325551 - 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}]

A325553 Number of compositions of n with distinct circular differences up to sign.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 7, 21, 31, 41, 87, 99, 191, 245, 381, 501, 735, 883, 1309, 1841, 2589, 3435, 4941, 6857, 9791, 13503, 19475, 27073, 37175, 52299, 72249, 100359, 139317, 190549, 256769, 355193, 471963, 644433, 858793, 1159161, 1530879, 2056073, 2711921

Offset: 0

Gus Wiseman, May 11 2019

nonn

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

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) = 21 compositions:
  (1)  (2)  (3)  (4)  (5)  (6)  (7)    (8)
                                (124)  (125)
                                (142)  (134)
                                (214)  (143)
                                (241)  (152)
                                (412)  (215)
                                (421)  (251)
                                       (314)
                                       (341)
                                       (413)
                                       (431)
                                       (512)
                                       (521)
                                       (1124)
                                       (1142)
                                       (1241)
                                       (1421)
                                       (2114)
                                       (2411)
                                       (4112)
                                       (4211)

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

Cf. A000079, A008965, A167606, A173258, A325324, A325349, A325545, A325549, A325551, A325552, A325553, A325556, A325558.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Abs[Differences[Append[#,First[#]]]]&]],{n,20}]

a(0) and a(26)-a(43) from Alois P. Heinz, Jan 28 2024

A325552 Number of compositions of n with distinct differences up to sign.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 12, 23, 38, 61, 78, 135, 194, 315, 454, 699, 982, 1495, 2102, 3085, 4406, 6583, 9048, 13117, 18540, 26399, 36484, 51885, 72498, 100031, 139342, 192621, 267068, 367631, 505954, 687153, 946412, 1283367, 1745974, 2356935, 3207554, 4311591, 5816404

Offset: 0

Gus Wiseman, May 11 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).
a(n) has the same parity as n for n > 0, since reversing a composition does not change whether or not it has this property, and the only valid symmetric compositions are (n) and (n/2,n/2), with the latter only existing for even n. - Charlie Neder, Jun 06 2019

			The differences of (1,2,1) are (1,-1), which are different but not up to sign, so (1,2,1) is not counted under a(4).
The a(1) = 1 through a(7) = 23 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)
       (11)  (12)  (13)   (14)   (15)   (16)
             (21)  (22)   (23)   (24)   (25)
                   (31)   (32)   (33)   (34)
                   (112)  (41)   (42)   (43)
                   (211)  (113)  (51)   (52)
                          (122)  (114)  (61)
                          (221)  (132)  (115)
                          (311)  (213)  (124)
                                 (231)  (133)
                                 (312)  (142)
                                 (411)  (214)
                                        (223)
                                        (241)
                                        (322)
                                        (331)
                                        (412)
                                        (421)
                                        (511)
                                        (1132)
                                        (2113)
                                        (2311)
                                        (3112)

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

Cf. A011782, A070211, A175342, A242882, A325325, A325368, A325404, A325545, A325551, A325553, A325555, A325557.

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

a(26)-a(42) from Alois P. Heinz, Jan 27 2024

A325591 Number of compositions of n with circular differences all equal to 1, 0, or -1.

Original entry on oeis.org

1, 2, 4, 6, 11, 15, 27, 43, 68, 116, 189, 311, 519, 860, 1433, 2380, 3968, 6613, 11018, 18374, 30633, 51089, 85208, 142113, 237055, 395409, 659576, 1100262, 1835382, 3061711, 5107445, 8520122, 14213135, 23710173, 39553138, 65982316, 110071459, 183620990, 306316328

Offset: 1

Gus Wiseman, May 12 2019

nonn

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

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(6) = 15 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (22)    (23)     (33)
             (21)   (112)   (32)     (222)
             (111)  (121)   (122)    (1122)
                    (211)   (212)    (1212)
                    (1111)  (221)    (1221)
                            (1112)   (2112)
                            (1121)   (2121)
                            (1211)   (2211)
                            (2111)   (11112)
                            (11111)  (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A309931.

Cf. A000079, A008965, A034297, A173258, A325351, A325551, A325553, A325558, A325589, A325590.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ[1,##]&@@Abs[DeleteCases[Differences[Append[#,First[#]]],0]]&]],{n,15}]

step(R,n,D)={matrix(n, n, i, j, if(i>j, sum(k=1, #D, my(s=D[k]); if(j>s && j+s<=n, R[i-j, j-s]))) )}
a(n)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)<=1), t=0); while(R, t+=R[n,k]; R=step(R,n,[0,1,-1])); t)} \\ Andrew Howroyd, Aug 23 2019

a(n) ~ c * d^n, where d = 1.66820206701846111636107... (see A034297), c = 0.65837031047271348106444... - Vaclav Kotesovec, Sep 21 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 23 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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A325553 Number of compositions of n with distinct circular differences up to sign.

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A325552 Number of compositions of n with distinct differences up to sign.

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A325591 Number of compositions of n with circular differences all equal to 1, 0, or -1.

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